The formula of antiderivative of ln x 2 contains integral sign, coefficient of integration and the function as sine. In question 2 is it possible to rewrite the equation to lnx * lnx instead of (lnx)^2 and integrate by parts? The problem I'm having when I try to use that method is that after I integrate lnx for the first time and substitute it back into the equation I get: ∫lnx * lnx dx = x(lnx)^2 - x - ∫(xlnx-x)/xHere's a classic integration problem involving a lnx and an x in the denominator. Trouble integrating in an efficient way. . Explanation: After setting dv = x2 ⋅ dx and u = lnx for using integration by parts, v = x3 3 and du = dx x. x . The Integral Calculator lets you calculate integrals and antiderivatives of functions online — for free! Our calculator allows you to check your solutions to calculus exercises. integral ( ln x )^2 d x. 9 to rewrite the result in terms of x . Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. #int u v' = uv - int u' v# #I = int ln x / x dx# #= int d/dx(ln x) ln x dx# #= ln x ln x - int ln x d/dx(ln x) dx# #= ln^2 x - int ln x/x dx# #implies I = ln^2 x - I + C#Secondly, I have attempted to find the n n th integral of ln(x) ln ( x) and this is what I found. You can also help support my channel by becoming a memberyour math problems using our free math solver with step-by-step solutions. Set up the integral to solve. For math, science, nutrition, history. It is often used to find the area underneath the graph of a function and the x-axis. (xln (x)) int e^xcos (x)dx int_{0}^{pi}sin(x)dx sum_{n=0}^{infty}frac{3}{2^n} Show More; Description. And dv = dx ⇒ v = x. Integrate functions involving logarithmic functions. Q. 12. The following is a list of integrals ( antiderivative functions) of logarithmic functions. In order to use sin(lnx)dx in dv, we'll need to be able to integrate dv. 3,505 2 2 gold badges 26 26 silver badges 37 37 bronze badgesIntegration of ln x^3 (Solution)Integration of ln x^3 (Solution)Integration of ln x^3 (Solution) - this video teaches us how to perform the Integration of ln. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform. Practice, practice, practice. You don't. In mathematics, the logarithmic integral function or integral logarithm li ( x) is a special function. That special log property: , 100 integrals: math for fun,In this case we take #u(x) = (lnx)^2# and #v(x) = x#, so that: #int (lnx)^2dx = x(lnx)^2-int 2xlnx(1/x)dx= x(lnx)^2-2int lnxdx# We solve this last integral again by parts: #int lnx = xlnx - int x*(1/x)dx = xlnx -int dx = xlnx -x+C#. [∫ sin (ln x),dx=xsin (ln x)−x cos (ln x)−∫ sin (ln x),dx. Calculate limits, integrals, derivatives and series step-by-step. Mar 26, 2008. ∫ ln ( x) 2 d x. Integral of ln x square formula. freak667. = u2 2 − 2u + lnu. Use these two facts: -lnx = ln x^(-1)=ln(1/x) e^lnu = u So, int x^(-lnx) dx = int 1/x dx=ln absx+C. Advanced Math Solutions – Integral Calculator, advanced trigonometric functions. Practice Makes Perfect. With introductory calculus in mind, ln | x | is defined as ∫ 1 x dx. 2. We could use substitution if we had the derivative of lnx as a factor, so we'll introduce it. Since u = lnx: ln(|u|) +C = ln(|lnx|) + C. Answer link. Line Equations Functions Arithmetic & Comp. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. d−n dx−n ln(x) = xn[ln(x) −∑n i=1 1 i] Γ(n + 1) d − n d x − n ln ( x) = x n [ ln ( x) − ∑ i = 1 n 1 i] Γ ( n + 1) Two problems about this formula, I'm unsure if it works fully, and I need it to work for non whole n n. I The graph of the natural logarithm. This is a common integral: ∫ 1 u du = ln(|u|) +C. Example. calculus-calculator int frac{1}{lnx}dx. In this video we will look at the integral of ln(3x) and learn how you can take care of the integration when you have a 3x appearing instead of just x. Where c is any constant involved, dx is the coefficient of integration and ∫ is. (ln (x))/x = 1/x ln (x) So we have the two functions; f. Solve problems from Pre. To evaluate this integral first we use the method of substitution and then we use integration by parts. Type in any integral to get the solution, steps and graphDerivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. en. Type in any integral to get the solution, steps and graphIntegration of 1/ln x (Solution)Integration of 1/ln x (Solution) - this video teaches us how to perform the Integration of 1/lnx (Solution)Check out other po. It is used to find the area under a curve by slicing it to small. 2) I Definition as an integral. ) Solving for (frac{dy. Taking the derivative of both sides: d dx u = d dx lnx. Then:🏼 - Integral of ln(x+sqrt(1+x^2)) - How to integrate it step by step using integration by parts!🔍 𝐀𝐫𝐞 𝐲𝐨𝐮 𝐥𝐨𝐨𝐤𝐢𝐧𝐠. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. how common is my name. Evaluate the integral, $$ int_{0}^{1} ln(x)ln(1-x),dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -zeta(2) $, but I don't think that it is best solution to this problem. en. Related Symbolab blog posts. However, I will show. . integral-calculator int lnx dx. The integral then becomes. In this case, this is not hard since the integral is dominated by $displaystyle int_0^t vert log(x) vert dx$. Type in any integral to get the solution, steps and graph. For the second integral: J = limx→+∞(−Li2(−ex) − x2 2 − (−π2 12)) J =. We see that on the right side. . . (xln (x)) int e^xcos (x)dx int_{0}^{pi}sin(x)dx sum_{n=0}^{infty}frac{3}{2^n} Show More; Description. Method 2. integral ln(x)dx. en. That special log property: , 100 integrals: math for fun,In this case we take #u(x) = (lnx)^2# and #v(x) = x#, so that: #int (lnx)^2dx = x(lnx)^2-int 2xlnx(1/x)dx= x(lnx)^2-2int lnxdx# We solve this last integral again by parts: #int lnx = xlnx - int x*(1/x)dx = xlnx -int dx = xlnx -x+C#. Tap for more steps. Then, we can apply rule (1). In the previous posts, we have covered three types of ordinary differential equations, (ODE). Thanks for the feedback. Practice, practice, practice. $int_0^1 ln x = x(ln x -1) |_0^1 = 1(ln 1 -1) - 0 (ln 0 -1)$ Is the second part of the calculation indeterminate or 0? What am I doing wrong? Thanks Joachim G. The function ln x is also known as the natural log of x. Sorted by: 12. Practice Makes Perfect. Solution: To find the integral of (ln x) 2, we will use the integration by parts method whose formula is ∫udv = uv - ∫vdu. $egingroup$ @Ron You directed me to your post, so I hope you're willing to answer some of my questions. ∫0 −∞tnetdt = (−1)n∫∞ 0 tne−pdp = (−1)nn! should your tn should be p on the right hand side after the −1). en. ∫logaxdx = x lna(lnx − 1) + C. integral int 1/ln x. Let u = lnx ⇒ du = 1 x dx. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. Calculate limits, integrals, derivatives and series step-by-step. Message received. Here it is: $$int ln(x)cos(x): mathrm{d}x$$ calculus; Share. In my view, writing the antiderivative as $ln|x|$ is a neat way of summarising two results in one, but it carries a serious risk of disguising what is really going on. Related Symbolab blog posts. Advanced Math Solutions – Integral Calculator, the complete guide. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…We begin by noting some obvious facts. . Simplify. Integrate by parts using the formula, where and . Though the 'at ease' part is usually the hardest! However, I do believe the greatest trait for any mathematician is luck. integral-calculator int e^{lnx}dx. The formula of integral of ln contains integral sign, coefficient of integration and the function as ln(x). Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. ∫ u d v = u v − ∫ v d u. The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. en. By the definition of the fundamental theorems of definite integrals, we can compute the definite integration of tan x between any two intervals. integral ln(x)dx. and use integration by parts. NEWO. Related Symbolab blog posts. Why is this? integration; definite-integrals; Share. (xln (x)) int e^xcos (x)dx int_{0}^{pi}sin(x)dx sum_{n=0}^{infty}frac{3}{2^n} Show More; Description. − ln ( tan x) cos x + ∫ 1 tan x 1 cos. Related Symbolab blog posts. Calculate limits, integrals, derivatives and series step-by-step. In the formula ∫ udv = ∫ ∫ u tan = ln tan. This calculus video tutorial explains how to find the integral of lnx using integration by parts. θ = sec −1 ( x a ) . Repeat as needed to reduce the exponent to zero, and one sees that the limit is zero. Type in any integral to get the solution, steps and graph. Let u = ln(x), dv 1 2dx, u = ln ( x), d v = 1 2 d x, so we have v = x 2, du = 1 xdx v = x 2, d u = 1 x d x. Related Symbolab blog posts. I The derivative and properties. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. . ∫lnxdx = xlnx −∫dx = xlnx −x = x(lnx − 1) Answer link. What follows is one way to proceed, assuming you take log to refer to the natural logarithm. Thanks. However, the integral you have requested is indefinite, which equals to the. And dv = dx ⇒ v = x. I'd like a hint toward how I could evaluate this definite integral. If (x>0) and (y=ln x), then (e^y=x. For each of the following integrals, use the result Z f′ f = ln|f|+cto determine the integral. In other words, the derivative of the natural logarithm of x is 1/x. Conic Sections Transformation. Hot Network Questions Do published research papers cite preprints?The definition of a definite integral is ; Given a function that is continuous on the interval [a, b] we divide the interval into n subintervals of equal width Δx, and from each interval choose a point xi. e: We often use Integration by Parts, when no other methods can solve the integral. If you can use the chain rule and the fact that the derivative of ex is ex and the fact that ln(x) is differentiable, then we have: d dxx = 1. en. Related Symbolab blog posts. Answer: the integral of ln (x) is xln (x) - x + C. Both of these are fairly simple integrals: I = xln(1 + x) − x + ln(1 +x) +C. Choose "Evaluate the Integral" from the topic selector and click to. Related Symbolab. en. Using integration by parts. What will be the integration of the following wrt x?Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. int frac{ln(x)}{x}dx. I = ∫ln xdx I = ∫ ln x d x. Now. Study Materials. = ln (x) x - x. Step 6. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin. Integral cosine in the complex plane. Solution of Integral frac {1} {lnx} rock. High School Math Solutions – Polynomial Long Division Calculator. Then we have dU = 1/x dx and V = x. It's going to be equal to pi times the indefinite integral of 1/u. I'd like a hint toward how I could evaluate this definite integral. It is denoted by ∫(ln x)dx. The Integration by Parts formula gives. This means . Important Notes on Antiderivative of ln x. This calculus video tutorial explains how to find the integral of lnx/x^2 using integration by parts. int lnx. 19: Axes for plotting the functions in Activity. . In mathematics, trigonometric integrals are a family of integrals involving trigonometric. Join this channel and unlock members-only perks. The first rule to know is that integrals and derivatives are opposites! Sometimes we can work out an integral, because we know a matching derivative. Type in any integral to get the solution, steps and graphSymbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more. $$ ∫ ln(x),dx,, $$ but integration by parts requires two. Integrate by parts using the formula, where and . en. Evaluate the indefinite integral: integral of ((ln x)^3)/(x) dx. The Integral Calculator solves an indefinite integral of a function. In this tutorial we shall derive the integral of Sqrt x lnx, and solve this problem with the help of the integration by parts method. en. Combine and . Examples. 1. Assume u = (ln x) 2, dv = dx, then du = (2/x) ln x dx and v = x. Free Online Integral Calculator allows you to solve. ∫x − 1dx = ln | x | + C. . integral-calculator. Find $int{arctan x},mathrm dx$ without substitution. ∫ cos log x dx = x cos log x + x sin log x − ∫. (xln (x)) int e^xcos (x)dx int_{0}^{pi}sin(x)dx sum_{n=0}^{infty}frac{3}{2^n} Show More; Description. Integral of ln(x^n) - How to integrate it by parts step by step!👋 Follow @integralsforyou on Instagram for a daily integral 😉📸 @integralsforyou = dx x. I nverse L ogarithm A lgebraic T rig E xponential. Evaluate the following indefinite integral: integral of ln 2 dx. Created by Sal Khan. Calculus 1 Final Exam Review:. The derivative of the logarithm ( ln x ) is ( frac{1}{x} ), but what is the antiderivative?This turns out to be a little trickier, and has to be done using a clever integration by parts. I would start with a u-substitution and separate the integral. Separating the summed integrals: = ∫ ln(x) x dx +∫dx. So consider the second function as 1 1. Integral of ln(ln(x))/(x*ln(x)) - How to integrate it step by step using the substitution method! Youtube: Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform. 3 Answers. I = ∫√xlnxdx ⇒ I = ∫lnx√xdx – – – (i) Here the first function is lnx and the second function is √x. Calculus 1 Final Exam Review: how to do the integral of ln(x), the natural logarithm function, by using integration by parts that you will learn in Calculus 2. I'm pretty sure that cut is the result of using the principal branch of $log(1+z^{2})$. Related Symbolab blog posts. Type in any integral to get the solution, steps and graphDerivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. Splitting the fraction ( x x = 1 ): = ∫ ( ln(x) x + 1)dx. Type in any integral to get the solution, steps and graph. . Related Symbolab blog posts. integral ln(x^2+1)x. In summary, the author is trying to find a solution for the homework statement ei (1,-1ln (x)). Compute answers using Wolfram's breakthrough technology &. ∫udv = uv − ∫vdu. It is relevant in problems of physics and has number theoretic significance. en. For x > 0, define the natural logarithm function by. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Let u = ln(x). Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Extended Keyboard. (xln (x)) int e^xcos (x)dx int_{0}^{pi}sin(x)dx sum_{n=0}^{infty}frac{3}{2^n} Show More; Description. 3. Where u and v are functions of x. Hence, the required integral is ∫ ln x d x = x ln x-x + C where C is a constant of integration . Integrate by parts using the formula, where and . en. calculus-calculator int left(lnx ight)^{2} en. Calculus 1 Final Exam Review:. Step 4. integral lnx^3. en. We. en. 3 Answers. 1 Answer. We have: ∫ (lnx)( 1 x2) dx = (lnx)( − 1 x) −∫ ( − 1 x)( 1 x) dx. 1. Save to Notebook! Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Integral of (lnx)^3/xIf you enjoyed this video please consider liking, sharing, and subscribing. Simplify. . The first rule to know is that integrals and derivatives are opposites! Sometimes we can work out an integral, because we know a matching derivative. 🏼 - Integral of x*ln(x) - How to integrate it step by step using integration by parts!🔍 𝐀𝐫𝐞 𝐲𝐨𝐮 𝐥𝐨𝐨𝐤𝐢𝐧𝐠 𝐟𝐨𝐫 𝐚. Related Symbolab blog posts. But since y = ln(x) y = ln ( x), we have ey =eln(x) = x e y = e ln ( x) = x and so our original. The image below shows the formula for the. Where c is any constant involved, dx is the coefficient of integration and. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Solve problems from Pre. I = ∫xsin(lnx) 1 x dx. Options. Save to Notebook! Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Type in any integral to get the solution, steps and graph. ∫ + ∞ 0 e − xlog(x)dx = Γ ′ (1) = Γ(1)ψ(1) = − γ follows from differentiation under the integral sign, as shown by Felix. ∫ (u − 1)2 u du. " Which is nothing but, integration by parts. Type in any integral to get the solution, steps and graph. Type in any integral to get the solution, steps and graph. Let: u = x+1 du = dx x = u - 1 => int (u-1)lnudu = int u lnudu - int lnudu With these two integrals in mind, we can do Integration by Parts. Integration Formulas Involving Logarithmic Functions. How to integrate ln(2x) by using the integration by parts methodThe integration technique involved here is to realize that ln(2x)dx is the same as ln(2x) tim. calculus-calculator int lnxdx. en. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. I Logarithmic differentiation. Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform. int ln(x)dx. Integral of ln(x^2)/x - How to integrate it step by step using the substitution method!👋 Follow @integralsforyou on Instagram for a daily integral 😉📸 @int. Integral(ln(x)) = xln(x) - Integral(x1/x) Awesome. The formula is given below. This calculus video tutorial explains how to find the integral of lnx/x using the u-substitution integration technique. Natural log of x is u-- we set that equal to natural log of x-- times du. integral-calculator. Infinite series can be very useful for computation and problem. Diffusion with spatially variable coefficient and source. I = 1 3[xlnx −∫dx] = 1 3 [xlnx − x] + c. Step 6. Solve problems from Pre. Generally speaking, "using ln(x) " as a rule or technique is unheard of. Let: #u = ln(x+1)# #du = 1/(x+1)dx# #dv = 1dx# #v = x# #uv - intvdu#. Why is the hyperbolic area of a horocycle non-convergent when the area of an ideal triangle is not?[int frac{ln{x}}{{x}^{2}} , dx] +. . I = ∫ln x ⋅ 1dx – – – (i) I = ∫ ln x ⋅ 1 d x – – – ( i)Integration of lnx. We would like to show you a description here but the site won’t allow us. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step evaluating the indefinite integral of lnx (the natural log of x) 1. Related Symbolab blog posts. What is the integration of ln x? Get the answer to this question and access a vast question bank that is tailored for students. Step 2. Related Symbolab blog posts. ∫x 11 t dt = − ∫1 x1 t dt, so in this case it is the negative of the area under the. Let u = x and. I mentioned: ∫ln(ex − 1)dx = −Li2(−ex) + C ∫ ln ( e x − 1) d x = − L i 2 ( − e x) + C. Related Symbolab blog posts. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. The function can be found by finding the indefinite integral of the derivative. li(x) =∫x 0 1 log(t) dt. If taking the definite integral of ln(x), you don't need the C. Parentheses are sometimes added for clarity, giving ln(x),. definite-integral-calculator int lnx dx. = 2∫lnx dx. 5. ∫lnxdx = xlnx − x + C = x(lnx − 1) + C. The antiderivative of ln x can be calculated using the method of integration by parts. Step 5. In general, it's always good to require some kind of proof or justification for the theorems you learn. It is relevant in problems of physics and has number theoretic significance. { int frac{lnx}{x^2}:dx } there was a -sign problem in my result , Althouh you guys are doing great job thanks khan. As we just saw, this is ln (x). Solutions. . We have now reached. Related Symbolab blog posts. Save to Notebook! Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. If you are studying maths, then you should learn the formula for Integration By Parts (IBP), and practice how to use it: # intu(dv)/dxdx = uv - intv(du)/dxdx #, or less formally # intudv=uv-intvdu # I was taught to remember the less. Integral of ln x square formula. We will be using integration by parts to find ∫lnxdx: ∫udv = uv − ∫vdu. Explanation: The integral of the product of two functions may be verbally given as, "First function into integral of the second minus integral of the derivative of the first into integral of the second. int 2ln(x)dx. Type in any integral to get the solution, steps and graph. This lesson covers the steps to solve, including application and solution, and takes a look at. 1 AnswerIntegral of ln(x)/x^4 - How to integrate it by parts step by step!👋 Follow @integralsforyou on Instagram for a daily integral 😉📸 @integralsforyou this video, we work through how to integrate ln^2(x). 18. I'm assuming that we have ∫lnx2 dx. So that implies du = dx / x, and v = x. integral-calculator. > <. Then du = 1 / (1 + x2)dx and v = x. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…The integral of xlnx is equal to (x 2 /2) lnx - x 2 /4 + C, where C is the constant of integration. We have the following identity for the integrals of inverse functions: ∫f − 1(x)dx = xf − 1(x) − F ∘ f − 1(x) + C, where F is the antiderivative of f, not f − 1. The solution is shown here (sometimes Wolfram Alpha has trouble computing integrals with lnx properly, so I gave the backwards check). So for the first integral: I = −Li2(−1) +Li2(0) = π2 12 I = − L i 2 ( − 1) + L i 2 ( 0) = π 2 12. Khan Academy | Free Online Courses, Lessons & Practice5 Answers. integral-calculator. Hence, the antiderivative of ln x square is given by ∫ [ln x] 2 dx = x [ln x] 2 - 2x ln x + 2x + K. Step 2: Click the blue arrow to submit. Advanced Math Solutions – Ordinary Differential Equations Calculator, Exact Differential Equations. int ln(lnx) dx = xln(lnx) - int 1/lnx dx Using integration by parts with u = ln(lnx) and dv = dx, we get int ln(lnx) dx = xln(lnx) - int 1/lnx dx The antiderivative of 1/lnx is. Check. Natural Logarithms (Sect. Since the original function involves lnx, we are justified in assuming that x>0 and writing: int x^(-lnx) dx =lnx+C. integrate (ln(x))^2. So the value of the whole integral is $0$. integral \int 1/ln x. etc. Euler-Minding summation algorithm for a regular continued fraction. 12. It is denoted by ∫ (cos (lnx))dx. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Related Symbolab blog. ln (abslnx)+C We have the integral: int1/ (xlnx)dx Use substitution. Let u = lnx ⇒ du = 1 x dx. =. integral-calculator. Integral of ln (x)/ (x+1) from 1 ot 0. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Related Symbolab blog posts. Similar questions. The integration of lnx l n x is of the form. Type in any integral to get the solution, steps and graph. 🏼 - Integral of ln(1+x^2) - How to integrate it step by step using integration by parts!🔍 𝐀𝐫𝐞 𝐲𝐨𝐮 𝐥𝐨𝐨𝐤𝐢𝐧𝐠 𝐟𝐨𝐫. 2. en. High School Math Solutions – Derivative Calculator, the Basics. en. Then repeat the integral, using algebra to simplify the integrand before integration. I = =∫ π 4 0 ln(1 + tan x)dx +∫ π 2 π 4 ln(1 + tan x) dxx→π 2−x 2∫ π 4 0 ln(1 + tan x)dx −∫ π 4 0 ln(tan x)dx = π 4ln 2 + G. ∫ ln ( x) 2 d x. I'm aware it's likely to be non elementary and I haven't found a way to evaluate it yet: $$int_0^infty ln( anh(x)),,mathrm{d}x$$ If you're curious where this came from, I was looking at an integral involving $ln(sin(x))$ and I thought of this one. The formula of cos (lnx) integral contains integral sign, coefficient of integration and the function as sine. The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. ∫uv dx = u∫v dx -∫ [ d u d x ∫v dx] dx. Thanks.