Questions tagged [indefinite-integrals]

Question about finding the primitives of a given function, whether or not elementary.

3,687 questions
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Integration of $f(x）= \exp(-ax-b\sin^{-1}(cx))$

Is there an analytical solution for the following integral? $$\int \exp(-ax-b\sin^{-1}(cx)) dx$$ Mathematica was unable to solve it, and I tried myself by integration by parts but without success.
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Integral of $\int x^{n-1}W(x)dx$

How to prove that : $$\int x^{n-1}W(x)dx = \frac {x^ne^{[-nW(x)]}[-nW(-x)]^{-n}[n\Gamma(n+1, -nW(x)- \Gamma(n+2, -nW(x))]} {n^2}$$ Where $W(x)$ is the Lambert-W function https://en.wikipedia.org/...
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What values of $a$ make $\frac{a-1}{3-a} x^{3-a} |_1^{\infty} - (\frac{a-1}{2-1})^2(x^{2-a} |_1^{\infty})^2$ finite?

What values of $a$ make $\frac{a-1}{3-a} x^{3-a} |_1^{\infty} - (\frac{a-1}{2-1})^2(x^{2-a} |_1^{\infty})^2$ finite? Is it a necessary and sufficient condition to simply make both the first and the ...
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Calculating value of Independently using Integral [duplicate]

Let $$I(a)= \int_0^\infty \frac{dx}{(1+x^a)(1+x^2)}$$ where $a\in\mathbb{R}$. Evaluate $I(a)$ and show that its value is independent of $a.$
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Bessel function limit

I was doing a physics problem and encountered with following limit of Bessel function : $\lim_{R\to\infty} R^2J_n(\lambda R)$ and $\lambda = \sqrt{\omega^2 - \frac{1}{R^2}}$ I got this limit from ...
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Integrate $\int\frac{\sin^{-1} (x)}{(1-x^2)^{3/4}} \,\mathrm d x$

Integrate $$\int\frac{\sin^{-1} (x)}{(1-x^2)^{\frac{3}{4}}} \,\mathrm d x$$ I have followed some steps from here, but am not able to solve this question. Any help would be appreciated. Update: After ...
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Partial fraction decomposition with two variables

While I was solving some exercises as training for a test I'm gonna have, I've noticed that some integrals that I have to solve, I don't know how to do them, and I have no explanation on my papers and ...
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Ways to simplify arctan() in integral results?

Lately when I was computing $$\int\frac{\mathrm{d}x}{1+\sqrt{1-2x-x^2}},$$I got the result$$-2\arctan{\frac{\sqrt{1-2x-x^2}-1}{x}}-\ln\left(1-\frac{\sqrt{1-2x-x^2}-1}{x}\right)+\mathbf{C}.$$However, ...
Is $e^{\int \frac {1}{x}dx}$ equal to $x$ or $|x|$?
I encountered this expression quite a lot of times as a part of the integrating factor while solving linear differential equations. $$e^{\int \frac {1}{x}dx}$$ For sometime, I wrote it as $x$, and ...