# Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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### Integral of the square of a floor function in an infinite sum

Consider the given integral: $$\int_{0}^{1} (\sum_{n=1}^{\infty}\frac{\lfloor 2^n x\rfloor}{3^n})^2 \,dx$$ Please suggest how to approach this problem.
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### If $f'(t)=x'(t)\ \theta(T-t)$, What is the accurate formulation of $f(t)$? (antiderivative over distributions/special_functions)

If $f'(t)=x'(t)\ \theta(T-t)$, What is the accurate formulation of $f(t)$? (antiderivative over distributions/special_functions) Maybe the question look simple, by I need to understand Why it is as so,...
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### Saddle Point analysis of integral

I want to make saddle point approximation of the integral $$I = \frac{1}{2 \pi i } \int_{\gamma - i \infty} ^{\gamma + i \infty} \frac{1}{z^6} e^{zt} dz$$ but as I see the function in the exponent i....
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### Help with Calculus Problem: Finding Maximum Value of Integral [closed]

I'm currently struggling with a calculus problem and would appreciate any help or guidance. The problem is as follows: The figure shows the graphs of the functions $F$ and $G$. For the functions $f$ ...
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### How to prove the limits about integral. [closed]

Let $f (x)$ be continuous on $[0,a](0<a<\pi)$.prove that $$\lim_{\lambda\to+\infty}\frac{1}{\lambda}\int_{0}^{a}f(x)\frac{\sin^2\lambda x}{\sin^2x}\mathrm{d}x = \frac{\pi}{2}f(0).$$ I thought ...
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### How to integrate? Please explain. [closed]

∫x^2/sin x How to integrate? Please explain.
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### How can I solve $\int_{0}^{\infty}{\frac{\log(x)}{x^2+4}}dx$ using $2\tan\theta$ as substitution for x? [duplicate]

$$\int_{0}^{\infty}{\frac{\log(x)}{x^2+4}}dx$$ -from the book Advanced Problems in Mathematics by Vikas Gupta So I tried to solve this integral by substituting $x=2\tan\theta$ so that we get the upper ...
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### In the last line for the answer, the two values should be added right? (31/631 - 241/631 should be 31/631 + 241/631) [closed]

Question Answer given I've attached a picture with question and the given answer
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### Calculating the maximum height a person can jump on the moon, given the maximum height they jumped on earth [closed]

A person on earth performs a high jump of 2.41m. Assuming the person is wearing an ideal spacesuit that does not adversely affect athletic performance, the acceleration due to gravity on the moon is 1....
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### Limit of $\sum_{t=0}^n a_n(t)$ for n tends to infinity

I am wondering about how to deal with a limit of the following form $$\lim_{n \rightarrow \infty} \sum_{t=0}^n a_n(t),$$ with the function $a_n(t)$ and the sum being dependent on $n$. Consider the ...
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### Numerical Integration: Why isn't Polynomial Approximation Working?

I have the following integration problem: $$\int_0^1{ -m f(x) \left(\int_0^x{f(u)} du \right)}^{m-1} dx$$ I attempted to approximate $\int_0^x{f(u)} du$ using Chebyshev interpolation, I took $n+1$...
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### Why does this Anti-derivative not give me the right answer? [closed]

I took the integral of 1/x Now when I take the definite integral of $f(x)$ from $0$ to $\pi$ I get $0$ since $\cos(0)=\sin(\pi)=\arcos(4)=0$ however this does not match with the graph and online ...
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### Proving that two integral are equal (changing variable of integration) [closed]

I want to prove that $$\frac{1}{\beta^2}\int_0^\beta d\tilde{\beta} \frac{n^2 \tilde{\beta}}{e^{\tilde{\beta} n}+1}=-\int_{n}^\infty dz \frac{z}{e^{\beta z}+1}$$ how do we change the variable of ...
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