Reduction formula doubt.

If In=∫(1a2+x2)ndx Prove that:In=x2a2(n−1)(a2+x2)(n−1)+2n−32(n−1)a2In−1 I used Ibp but couldn’t get such a relation. Please help me. Also, please do not use induction. Answer Hint: By parts, setting u=1(a2+x2)n−1, dv=dx, whence du=−2(n−1)x(a2+x2)ndx,v=x One obtains then In−1=x(a2+x2)n−1+2(n−1)∫x2(a2+x2)ndx Note that, writing x2=a2+x2−a2, the integral is equal to In−1−a2In. AttributionSource : Link , Question Author : Aditya Kumar , … Read more

Prove that $(2n+1)k_{n+1}=(2n+1)k_{n}+\cos^{2n+1} (x)$

Given that $$k_n=\int \frac{\cos^{2n} (x)}{\sin (x)} dx$$ Prove that $$(2n+1)k_{n+1}=(2n+1)k_{n}+\cos^{2n+1} (x)$$ I have tried to prove this is true by differentiating both sides with product rule: $$2k_{n+1}+\frac{\cos^{2n+1} (x)}{\sin (x)}(2n+1)=2k_n+\frac{\cos^{2n} (x)}{\sin (x)}(2n+1)+(2n+1)\cos^{2n} (x) \sin (x)$$ I am stuck here as I met a dead end upon grouping and expanding. Please help. Thank you in advance!! Answer … Read more

Two apparently different evaluations of $\int \frac{x-1}{9x^2-18x+17}dx$

Evaluate the indefinite integral $$\int \frac{x-1}{9x^2-18x+17} \, dx .$$ This is an exercise from a book I’m studying. It gives the answer as: $$\ln(9x^2 -18x+17)^\frac{1}{18} +C .$$ This is an easy integral. You just notice that the numerator is the derivative of the denominator. But I didn’t notice exactly that at first, so I solved … Read more

How to find \int e^{2\cos 2x}dx\int e^{2\cos 2x}dx?

I have found a function. I’m unable to find its integral. I have searched for it in calculus books but couldn’t find answer. I just want a hint. \int e^{2\cos 2x}dx. Also about the function… \int e^{x^2}dx. Or any function with complicated power. Answer Using \displaystyle e^x = 1+\frac{x}{1!}+\frac{x^2}{2!}+………..\infty So \displaystyle \int e^{x^2}dx = \int … Read more

Clever way of calculating the integral $ \int \frac{dt}{t^2\sqrt{t-2} } $

$$ \int \frac{\text{d}t}{t^2\sqrt{t-2} } $$ I know it can be calculated using somewhat complicated substitutions, but is there possibly some clever way of solving that type of integral? I don’t even expect full solution, just ideas. Answer Substitute $t=2\tan^2(u)+2$: $$ \begin{align} \int\frac{\mathrm{d}t}{t^2\sqrt{t-2}} &=\frac1{\sqrt2}\int\cos^2(u)\,\mathrm{d}u\\ &=\frac1{2\sqrt2}\int(1+\cos(2u))\,\mathrm{d}u\\ &=\frac1{2\sqrt2}\left(u+\frac12\sin(2u)\right)+C\\ &=\frac1{2\sqrt2}\left(u+\frac{\tan(u)}{1+\tan^2(u)}\right)+C\\ &=\frac1{2\sqrt2}\arctan\left(\sqrt{\frac{t-2}2}\right)+\frac12\frac{\sqrt{t-2}}t+C\\ \end{align} $$ AttributionSource : Link , Question Author … Read more

Solve partial integral ∫x2ln(1−x1+x)dx∫ x^2 \ln \left(\frac{1 – x}{1 + x} \right)dx

Hello there this is my first post on this stack exchange community. I joined this community because i’m on my 2nd year of studying software engineering and as one of this year subjects is MA202, where we learn all about the joys of integrals. So this is my problem, in the title, i know it’s … Read more

What’s a method for computing the indefinite integral ∫dz(a2+z2)3/2\int \dfrac{dz}{(a^2 + z^2)^{3/2}}?

This integral occurs in EMFT when computing ¯E due to an infinite line, uniform charge distribution. I’m trying to figure out the formula for ∫dz(a2+z2)3/2, using integration by parts but fail. Choosing dv=(a2+z2)−3/2dz leaves us within the same boat we started. Choosing u=(a2+z2)−3/2, dv=dz, gives: du=−(3/2)(a2+z2)−5/2dx so that ∫udv=uv−∫vdu=z(a2+z2)−3/2−∫2z2(a2+z2)5/2dz Leaving me with a mess. What’s the … Read more

How to integrate xex−1\frac{x}{e^x – 1} w.r.t. x?

A friend of mine and I wanted to solve the following indefinite integral but got stuck: ∫xex−1dx. My approach: Let I=∫xex−1dx.⟹I=x∫dxex−1−∫(∫dxex−1)dx. Now, let I2=∫dxex−1. Also, let z=ex⟹dx=dzz. Then, I2=∫dzz(z−1)⟹I2=∫dzz−1−∫dzz⟹I2=ln(ex−1)−x. Substituting the value of I2 in I, we get, I=x[ln(ex−1)−x]+x22−∫ln(ex−1)dx. I got stuck right here. Is it possible to proceed further? Answer The antiderivative is not … Read more

Indefinite integral – tricky [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 4 years ago. Improve this question I am trying to solve this integral which looks simple but for some reason I can’t reach the final result. \int\ln(1+x^4) dx … Read more