# Tagged Questions

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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### How to integrate logarithm and power function?

I am trying to solve the following integral $$\int_{0}^{1}(\ln(1+x))^2 x^{a-1}\,dx ; a>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
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### I have a problem with solving this integral by partial fraction

I have an example like this : $\int \frac{1}{x^{4}}dx$ = $\int x^{-4}dx$ = $\frac{1}{-4 + 1}x^{-4 + 1}+C$ = $\frac{1}{-3}x^{-3}+C$ What if this? $\int \frac{1}{2x-10}dx$ = ?
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### Change of variables in $\int_{\Omega}(x+y+z)^2xyz \: dxdydz$

I have the triple integral $\int_{\Omega}(x+y+z)^2xyz \: dxdydz$ , with $\Omega=\lbrace (x,y,z)\in \mathbb{R}^3:0\leq x+y+z\leq 1 , \:1\geq x\geq0, \: \: 1\geq y\geq 0, \: \:1\geq z\geq 0 \rbrace$. I ...
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### How to evaluate the following integral, $\int\frac{x \, dx}{x^2+2x+17}$?

I am new to integration. This function is kinda tricky for me : $$\int\frac{x \, dx}{x^2+2x+17}$$ I came up with following three approaches: Partial fraction decomposition, but I can't factor the ...
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### Integration on complex spheres and Gamma function

I'm studying special functions, especially Jacobi functions, related to the rank one groups ($U(1, n; \mathbb{F})$ where $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$, the skew-field of quaternions), ...
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### Evaluate the integral $\int \frac{1}{x^3\sqrt{x^2+x-1}}\mathrm dx$

What substitution to use in this integral? I tried to factorize $x^2+x-1$ and use a substitution $u=\sqrt{\frac{a(x-\alpha)}{x-\beta}}$.
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### Integration by Substitution Rearranging

I have to show that the following the integral in terms of x can be rewritten as follows: $$\int_{-1}^1\frac{\sqrt{1-x^2}}{1+x^2}dx = \int_{u_2=}^{u_1=}\frac1{1+cos^2u}du-\pi$$ I have used the ...
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### Solve the integral $\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx$

To find the Variance of a Wiener Process, $Var[W(t)]$, I have to compute the integral $$Var[W(t)]=\dots=\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx=\dots=t.$$ I've ...
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### Solve $\int{\frac{7x^2 + 1}{(x+1)(x-1)(x+3)}}\,dx$ [closed]

I don’t know how to solve this integral: $$\int{\frac{7x^2 + 1}{(x+1)(x-1)(x+3)}}\,dx$$ I know this is a rational integral but I don’t know how to write it in a different way
Suppose $X$ and $Y$ are independent, standard normal random variables. I'm trying to compute the probability of the event $$\{X \leq x, Y \leq kX\}$$ where $k$ is a positive constant. The ...