Linked Questions

Find $\displaystyle \int x^ne^xdx$. I haven't been able to figure this out. Do I just keep repeatedly applying integration by parts?

Hello could any one tell me some unusual or advanced integration techniques, I am already familiar with the standard ones like u-substitution, integration by parts, trig substitution, partial ...

When integrating a real rational fraction, you first break it into partial fractions. You then end up with fractions with linear denominators $\frac{A}{(x-b)^n}$, which are easy. You also end up with ...

I am trying to integrate $\int \frac 1 {(x^2+a^2)^2} \ dx$. The only thing that I can think to try is substitution, $u=x^2+a^2$ so that $\frac{du}{dx}=2x \Rightarrow du = 2x\ dx = 2\sqrt{u-a^2}\ du$ ...

I'm trying to use a trig substitution but I'm stuck. Here's what I did so far: $$\int \frac{dx}{(a^2 + x^2)^2}$$ Let $x = a\sin \theta, dx = a\cos \theta d\theta$ $$\int \frac{a cos\theta d\theta}{...

Evaluate $$\int\frac{dx}{(1+x^2)^4}$$ Now I did solve it, but I used the mentioned substitution and after a lot of converting into double angles, I did it. But, it doesn't look like a good approach. ...

I would like to find the anti-derivative $$\displaystyle \int \dfrac{dx}{(x^2 + a^2)^3 }$$ My attempt: By substitution: $ x = a \tan(\theta) \Rightarrow dx = a \sec^2(\theta) d\theta$ Then the ...

Our calculus book covers partial fractions but not trig substitution, so I would like to find out the most elementary way to evaluate $$\displaystyle\int\frac{1}{(t^2+25)^2}\;dt$$ without using ...

On the article Lesser known integration tricks, I found this: \begin{align*} J &= \int (1 + 2x^2) e^{x^2} \mathrm{d}x \\ &= \int 2x^2e^{x^2}\mathrm{d}x + \int e^{x^2} \mathrm{d}x \, \\ ...

How to find $$\displaystyle \int_0^\infty \frac{1}{(1+x^2)^n}dx\quad ?$$ For $n=1$, we have $(\arctan x) |_0^\infty = \frac{\pi}{2}.$ I tried to integrate by parts to get recurrent formula: $$ \...

How do I integrate $\frac{1}{(x^2+1)^2}$? I've tried to use the fact that $\int \frac{1}{(x^2+1)}=Arctan(x)$ but I don't know how to. I think it's by parts. Tried using $u'=(x^2+1)^{-1}$ and $v=(x^2+...

How to calculate the following integral without using wolfram mathematica: $$ \int\frac{dx}{(x^2-4x+5)^2} $$

I worked differential linear equation and at end of equation I got this integral.Can someone give me a hint to do this: $$\int \frac{1}{\left(u^2+1\right)^2}du$$

I'm compiling a list of interesting definite integrals for an upcoming blog post, and I thought that the math SE community might have a few interesting problems to offer. I am especially interested in ...

Evaluate $$ \int_0^\infty\frac{x\log x}{(1+x^2)^2}dx $$ $$ \int\frac{x}{(1+x^2)^2}dx=\frac{-1}{2(1+x^2)}\\ \int_0^\infty\frac{x\log x}{(1+x^2)^2}dx= \bigg[\log x\frac{-1}{2(1+x^2)}\bigg]_0^\infty-\...