# Questions tagged [indefinite-integrals]

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

3,690 questions
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### Evaluating $\displaystyle\int \frac{\cos x}{\sin^3x+\sin x}dx$

Given the function $$g(x)=\frac{\cos x}{\sin^3x+\sin x}$$, by letting $u=\sin x$, show that $$\int g(x) dx=\int\left(\frac{A}{u}+\frac{Bu+C}{u^2+1}\right)du$$ where $A,B$ and $C$ are constants. Hence, ...
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### Unable to solve $\int \frac{x + \sqrt{2}}{x^2 + \sqrt{2} x + 1} dx$?

This comes from a bigger problem :- $$\text{Evaluate } \int\frac{dx}{1+x^4}$$ After making $\int \frac {dx}{1+x^4} = \frac{dx}{(1+x^2)^2 - (\sqrt{2}x)^2}$ and then applying partial fraction ...
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### Solve integral $\int_0^\infty \sqrt{x}e^{-\sqrt[3]{x}} dx$

$$\int_0^\infty \sqrt{x}e^{-\sqrt[3]{x}} dx$$ I have tried solving it by substituting $x=u^2$ but couldn't solve it.
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### Definite Integral of $\int_0^1\frac{dx}{\sqrt {x(1-x)}}$

We have to calculate value of the following integral : $$\int_0^1\cfrac{dx}{\sqrt {x(1-x)}} \qquad \qquad (2)$$ What i've done for (2) : \begin{align} & = \int_0^1\cfrac{dx}{\sqrt {x(1-x)}} \\ &...
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### Suppose that the average value on all intervals $[a,b]$ is equal to $f((a+b)/2)$. Prove that $f''(x) = 0$ for all $x \in \mathbb{R}$
I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on ...