Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

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Showing that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges (Baby Rudin Exercise 6.9)

Motivated by Baby Rudin Exercise 6.9 I need to show that $\int_0^\infty \frac{|\cos x|}{1+x} \, dx$ diverges. My attempt: $\frac{|\cos x|}{1+x} \geq \frac{\cos^2 x}{1+x}$, and then $\int_0^\infty \frac{\cos^2 x}{1+x} \, dx + \int_0^\infty…
MT_
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Change of variables in $\iiiint$ or $\idotsint$?

I apologize if this is a question that has been asked before, but I have seen that it is possible to change variables in single, double and triple integrals. Now what about quadruple integrals, or $n$-integrals? I've Googled a lot and cannot seem to…
bjd2385
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Evaluation of $\int_0^a \frac {\cos[(\alpha-\beta)x]}{\sin\beta x} dx$.

Can anyone explain me how to evaluate the following integral in detail. $$\int_0^a \frac {\cos[(\alpha-\beta)x]}{\sin\beta x} dx$$ where $a\in[0,2\pi]$, $\alpha, \beta\in\mathbb R$ and $\alpha\neq \beta$. Maybe it can't be done analytically, so…
Mark
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Find the integral $\int \frac{\left(\sin\left(2 \:x\right)\right)^2}{\left(\sin^3x+\cos^3x\right)^2}$

That's decided a similar example. This I do not know how to solve. Help me please. My integral: $$\int \frac{\left(sin\left(2 \:x\right)\right)^2}{\left(sin^3x+cos^3x\right)^2}$$
andre1
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How to find the integral $\int_{-\infty}^{\infty}\frac{dx}{1+ae^{bx^2}}$

Could somebody tell me how to find the integral $$\int_{-\infty}^{\infty}\frac{dx}{1+ae^{bx^2}}$$ for constants $a$ and $b$? Thanks!
xianlong wei
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Inequality of integration

Let f be a positive continuous function on [0,1]. Set $A=\int_0^1 fdx$ Prove that $\sqrt{1+A^2}\le\int_0^1\sqrt{1+f^2}dx\le1+A$ I proved R.H.S inequality. So my question is L.H.S inequality. I guess ... The cauchy-Schwarz‘s inequality is…
user128766
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Common integral question

I was asked to find $$\int \sin^5x \, \mathrm dx$$ only in form of $\cos x$. I solved it using recursion, but I was getting something like $\cos^5x+ \cos^3x+\cos x$(obviously with some coefficients), but the options also had $\cos^4x+\cos^2x$ but I…
mathgeek
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The integral of $\sqrt{1+x^4}$?

For the integral $$ I=\int \sqrt{1+x^4}dx,$$ Mathematica can give a result like this: $$I=\frac{x^5-2 \sqrt[4]{-1} \sqrt{x^4+1} F\left(\left.i \sinh ^{-1}\left(\sqrt[4]{-1} x\right)\right|-1\right)+x}{3 \sqrt{x^4+1}}$$ where $F(a|b)$ is the…
Mark_Phys
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How to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$?

I'm trying to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$ by using the formula: $$ \int udv = uv -\int vdu $$ I supposed that $u=x$ s.t $du=dx$, and also that $dv=\frac{x}{(x^2+1)^2}dx$, but I couldn't calculate the last integral. what is the tick…
Firas Abd El Gani
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Does this integral limit exist?

let $f,g\in L^2(\mathbb R)$ and note the sequence $x_n=|\int_\mathbb R f(x+n)g(x)dx|$. Does $x_n$ converge as a sequence in $\mathbb R$? My guess would be yes, and that the limit is $0$, because in the edges $f$ should "tend" to zero, otherwise it…
Sanjo
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Antiderivative rational function

I'm searching the antiderivative of rational functions : 1)$\int \frac {1+x}{\sqrt{2x+1}} dx$ For this one we have $t=\sqrt{2x+1}$ then $dt=\frac {1}{\sqrt {2x+2}} dx$ but then I do not see the way to compute the antiderivative. Same thing for…
Tom75
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What are good applications of the generalized Integration by Parts formula?

I came across the generalized Integration by Parts today: Let $f \left({x}\right), g \left({x}\right)$ be real $n$ times differentiable functions with continuous $n$-th derivatives. Then: $$ \int f^{(n)}(x)g(x) dx = \sum_{j= 0}^{n-1}(-1)^j…
Taladris
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relationship between $\sin$ and $\cos$ integrals

I am trying to solve a physics problem and I end up with the following two equations. $\int\limits_{0}^{T}\sin\theta(t)dt = a$ and $\int\limits_{0}^{T}\cos\theta(t)dt = b$ The exact dependence of $\theta(t)$ with $t$ is not exactly known but it can…
shailesh mishra
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Integral with Limit and Parameters

I am stuck on the following problem evaluating an integral with parameters, where the parameter has a limit: $$\lim \limits_{x \to \infty} \int_0^\infty \sin \left(e^{xt}\right)\,dt$$ I know that in some cases you can differentiate what is contained…
Freddie
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How to solve $\int \tan^2(x) \sec (x) \ dx$?

How should one solve the following integral? $$\int \tan^2(x) \sec (x) \ dx$$ I can't think of any substitutions to be made involving $\tan^2(x)=\sec^2 (x)-1$ or $\sec^2(x)=\tan^2(x)+1$, which is how I've been solving most of the similar problems in…
Asker
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