# 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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### Calculating a definite improper integral - part 2

This question is regarding my previous question on the site. I want to calculate the following integral $$\int_0^{\infty} \frac{x^p(1+x)^{-p-1}}{r+sx}dx, p,r,s\in \mathbb{R}^+$$ according to the ...
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### Can $\int e^{-|x|^a}\,dx$ where 0<a<1 be given an analytical solution?

As the title suggests. Are there any solutions to solving this integral analytically or by using a power series? $\int e^{-|x|^a}dx$ where $0<a<1$? Any help, suggestions, or leads given are ...
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### A lengthy integral $\frac{\int_{0}^{a} x^4 \sqrt{a^2 - x^2} \, dx}{\int_{0}^{a} x^2 \sqrt{a^2 - x^2} \, dx}$

$$\frac{\int_{0}^{a} x^4 \sqrt{a^2 - x^2} \ dx}{\int_{0}^{a} x^2 \sqrt{a^2 - x^2} \ dx}$$ I tried solving this using trigonometric substitution, and it transforms into a very lengthy expression. Is ...
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### Derivative of a time-reversed unit step function?

I understand the idea of the unit step function and the Dirac delta, and that the delta can be seen as the "generalised" derivative of the former. But what about if we had to differentiate a ...
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### If an antiderivative exists, will it always be the indefinite integral?

Let $f$ be a function $[a,b] \to \mathbb{R}$. I define antiderivative as a function $F$ such that $F' = f$, and I define indefinite integral of $f$ as the function $\int_0^x f(t) dt$. The Fundamental ...
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### Evaluate $\int_{\ -\infty\ }^{\infty\ }\frac{1+x^3}{1+x^5}dx+2\int_{\ -1}^1\ \frac{x^5-x^3}{1+x^5}dx$ [closed]

Evaluate $\int_{\ -\infty\ }^{\infty\ }\frac{1+x^3}{1+x^5}dx+2\int_{\ -1}^1\ \frac{x^5-x^3}{1+x^5}dx$. One thing that I have noticed is that it's not possible to evaluate the integrals separately (at ...
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### Understanding the invariance of this infinite series geometrically

Recently I worked through a computation that $L=1/4$. I couldn't help but notice $L_{\rho}$ seems to be invariant under "re-scalings" of the the underlying arithmetic function. What I mean ...
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### Integral of $\int_{a}^{b} \sqrt{\left(\frac{1}{x}-1\right)}dx$

How can we find the integral $\int_{a}^{b} \sqrt{\left(\frac{1}{x}-1\right)}dx$, where $a, b$ both are positive. I tried to use method of substitution, but it is not working. I am curious to know, is ...
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A rectangular cistern has its top $10$ ft. below ground level. Its dimensions are $20$ ft. long x $12$ ft. wide, and $10$ ft. high. It is filled with water; therefore, its volume is $12 \times 20 \... • 31 -2 votes 3 answers 103 views ### Notion of an integral as a summation [closed] While analyzing the foundation of calculus I am finding that the notion of an integral is a special form of summation of differentials, and an indefinite integral is also an integral with limits$0$... • 103 6 votes 1 answer 145 views ### Double integral$\int\limits_{y = 1}^\infty {\int\limits_{x = 1}^y {\frac{1}{{xy{{\left( {y - x} \right)}^{\frac{1}{5}}}}}dxdy} } \$
I am trying to evaluate this integral $$I=\displaystyle\int\limits_{y = 1}^\infty {\int\limits_{x = 1}^y {\frac{1}{{xy{{\left( {y - x} \right)}^{\frac{1}{5}}}}}dxdy} }$$ By using CAS checks, the ...