# Questions tagged [indefinite-integrals]

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

3,693 questions
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### How do I find $\int e^{x \sin x +\cos x}(\frac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos ^2 x})\,dx$?

How do I find $$\int e^{x \sin x +\cos x}\left(\frac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos ^2 x}\right) dx\quad?$$ I tried to put $e^{x \sin x +\cos x}$ as some t and write the expression in ...
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### Does $\int \frac{e^x(x-1)}{1+xe^x}dx$ have any closed form?

It's simple that $$\int \frac{e^x(x+1)}{1+xe^x}dx=\ln(1+xe^x)+C,$$ But what if $$\int \frac{e^x(x-1)}{1+xe^x}dx?$$
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### Indefinite integral through factorization

So I have $$\int \frac{1}{(x^2+1)^2}dx$$ And the professor does some magic I'm confused. what's with the derivative? I solved the integral via substitution but I'm curious how this works, so I can ...
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### Evaluation of the integral $\int \frac{x^{\frac{1}{3}}}{1+x^3 } dx$ [closed]

I'm looking to solve this integral right here: $\int \frac{x^{\frac{1}{3}}}{1+x^3 } dx$ I would like to know what approaches I could take to solve this using complex analysis.
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### Integral $\int_\sigma^\infty r^2 {e^{-A/r^6}} dr$ [closed]

Below integral can be calculated by using taylor expansion for the $e^{-A/r^6}$ term. I want to know how to solve this integral analytically? $$\int_\sigma^\infty r^2 {e^{-A/r^6}} dr$$ Hint: I ...
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### Integrate $\int \frac {\sin (2x)}{(\sin x+\cos x)^2}\,dx$

Integrate $$\int \frac {\sin (2x)}{(\sin x+\cos x)^2} \,dx$$ My Attempt: $$=\int \frac {\sin (2x)}{(\sin x + \cos x)^2} \,dx$$ $$=\int \frac {2\sin x \cos x}{(\sin x+ \cos x)^2} \,dx$$ Dividing the ...
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### Compute $\int_{t-2T}^T t-\tau\cdot 1 \;d\tau = [t\tau - \frac{\tau^2}{2}]_{t-2T}^{T}$

Compute $\displaystyle\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau$ \begin{align} \displaystyle&\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau \\ &=\left[t\tau - \frac{\tau^2}{2}\right]_{t-2T}^{T}\\...
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### u-substitution yields a different answer

When I compute the indefinite integral of ln(x + x^2), I get 2 answers from 2 different methods. First method: integration by parts => u-substitution Answer = xln(x + x^2) - 2 (x + 1) + ln l x + 1 l ...
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### CAS integrals of discontinuous functions

Background This post is motivated by my interest in the performance of symbolic integrators in computer algebra systems (CAS's), such as Mathematica (MMA). I've found that, when an integrand has ...
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### Integrating $\int\tan^3(x)\,dx$ in two different ways gives two different answers

I was trying to find the antiderivative of a function $$\int \tan^3(x)\,dx$$ However, due to substitution differences, my book has a answer of $$\frac12\tan^2(x)+\ln(\cos x)+C$$ while I got an ...
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### How to solve given integral? [closed]

I need it for my Fouier series's coefficent. $$\int_{-π}^{π}\left| x\right| \cos{5x} \, dx$$