# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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### Why can't Wolfram Alpha calculate $\int_0^{2\pi}\sqrt{(a-\cos\theta)^2+\sin^2\theta}\ d\theta$?

In this answer to How is the average distance between 2 objects orbiting around a third object calculated? I had to integrate $$\int_0^{2 \pi}\sqrt{(a-\cos \theta)^2 + \sin^2 \theta} \ d\theta.$$ I ...
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### Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$

I'm interested in the following integral: $$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$ where $K(z)$ is the complete elliptic integral of the 1ˢᵗ ...
599 views

### How can I prove $\int_{0}^{1} \frac {x-1}{\log(x) (1+x^3)}dx=\frac {\log3}{2}$

Question:- Prove that $$\int_0^1 \frac {x-1}{\log(x) (1+x^3)} \, dx = \frac {\log(3)}{2}$$ I saw this problem as an comment on a youtube video few hours ago but I don't know how to prove this one as ...
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### Is there a known way to get the average length of a ray bounded by a cone and a box?

I am struggling to come up with a solution to the following problem: Imagine a cone defined by angle $\theta$ that passes through a rectangular prism as is shown below. How would one determine the ...
188 views

### Integral area of $\int_{1}^{e}\frac{3\ln\left(x\right)}{x \sqrt{\ln^{2}\left(x+1\right)}}dx$

Could someone please help me to solve this integral area: $\int_{1}^{e}\frac{3\ln\left(x\right)}{x\cdot\sqrt{\ln^{2}\left(x+1\right)}}dx$ I have no clue how to do it and I tried to do substitution and ...
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### Integrate and find closed form for $\int_0^\infty\frac{\sin x^n}{n\pi}dx$

I’m working on a practice exam for my analysis class, and I was asked to find a general form for $$\int_0^\infty\frac{\sin x^n}{n\pi}dx$$ When I first looked at this, my mind instantly went to the ...
### Show that $\int_0^1K^2(k)dk=\frac12\int_0^1K'^2(k)dk$
By switching integrals in double integral, I showed that $$\int_0^1K(k)dk=\int_0^1K'(k)dk=2G$$ where $K(k)$ is complete elliptic integral of the first kind and $K'(k)=K(\sqrt{1-k^2})$ is its ...