# Tagged Questions

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### Find the indefinite integral $\int\frac{(x+1)e^x}{x(1+xe^x)}dx$

Find the indefinite integral $$\int\frac{(x+1)e^x}{x(1+xe^x)}dx$$ I feel like this function does not have an anti-derivative in the form of elementary functions.
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### When we can change $\int$ and $\sum$ for indefinite integral?

I know, for example, that if the series $\displaystyle\sum_{n=1}^{\infty}f_n(x)$ consisting of integrable functions on a closed interval $[a, b] \subset \mathbb{R}$ converges uniformly on that closed ...
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### Let $f:[0,1]→\mathbb{R}$with $f′(x)$continuous. It is known that $\int_{0}^{1} f(x)dx=0$.

Let $f:[0,1]→\mathbb{R}$ with $f'(x)$ continuous. It is known that $∫_0^1 f(x) dx=0$. Prove that $∀α∈[0,1]$, $$|\int_{0}^{\alpha} f(x) dx |≤ \frac{1}{8} sup_{(0≤x≤1)}|f'(x) |$$ My answer so far ...
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### Evaluate $\int\sin(\sin x)~dx$

I was skimming the virtual pages here and noticed a limit that made me wonder the following question: is there any nice way to evaluate the indefinite integral below? $$\int\sin(\sin x)~dx$$ Perhaps ...
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### Derivative of the indefinite integral and Lebesgue point

Give an example where the derivative of the indefinite integral exists at point that are not Lebesgue points.
### Evaluating $\int \cos(x) \sqrt{\sin(2 x)} dx$
Evaluate the following indefinite integral: $$\int \cos(x) \sqrt{\sin(2 x)} dx$$ Only hint I have is from W|A that expresses the integral in terms of a hypergeometric function and it looks ...
### Evaluating: $\int \frac{t}{\cos{t}} dt$
How would you evaluate the following indefinite integral? In fact, I did evaluate $\int \frac{\cos{t}}{t} dt$ by parametric integration and then I thought of this variant. \int \frac{t}{\cos{t}} ...