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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How to find $\int \frac{\mathrm dx}{\sqrt{x}+\sqrt[3]{x}}$

Find $$\int \frac{\mathrm dx}{\sqrt{x}+\sqrt[3]{x}}$$ I substituted $t = \sqrt x$ so $x = t^2$ and $\mathrm dx = 2t \mathrm dt$. I got to the $$ 2\int \frac{dt}{1+t^{-\frac13}} $$ I'm not sure, if that is right. I still do lots of mistakes, but…
user50222
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Calculate double integral by changing the order of integration

I'm trying to do my mathematical analysis homework but I have faced one struggle. $$\iint_{\omega} y^2\,dx\,dy$$ where $\omega$ is bounded by the axis of abscissas and the first arch of the cycloid $$ \left\{ \begin{array}{c} x=a(t-\sin(t)) \\…
Karagum
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How do show $\int_0^1 \int_0^{1-x} \arctan\left(\sqrt{y/x}\right)/\sqrt{xy} \mathrm d y \mathrm = \pi^2/4$.

Maple shows that $$ \int_0^1 \int_0^{1-x} \arctan\left(\sqrt{y/x}\right)/\sqrt{xy} \, \mathrm d y \,\mathrm d x = \pi^2/4. $$ It looks simple but seems rather tedious to do compute manually. Is there easy proof for this? BTW, converting to polar…
NonalcoholicBeer
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Evaluate $\int_1^4\frac{dx}{x^2+x+1}$

I've to evaluate the integral $$\int_1^4 \frac{dx}{x^2+x+1}$$ but I can't find the answer. I checked with Wolfram Alpha but I still don't fully understand. Could you please explain the steps to me? I think I should use arctan in my answer.
Bob
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Finding the area

I was given the problem: find the area of the region bounded by $y=1/x$, $y=x^2$, $y=0$, and $x=e$. My approach was to break it up into two integrals, $\displaystyle \int_0^1 (x^2-0)\,dx$ and $\displaystyle…
Ryan
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How to evaluate $\int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy$

How to evaluate $$F(\Lambda)=\int_0^{\infty}\int_0^{\infty}\left(\frac{e^{-\Lambda x}-e^{-\Lambda y}}{x-y}\right)^2 \,dx\,dy$$ where $\Lambda$ is a positive real number? I tried evaluating the innermost integral…
Kanak Dhotre
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Integration questions on $\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$ and $\int \frac{x^4}{x^4+5x^2+4} \, dx$

I would appreciate any hints on how to solve the following integration problems, they are my homework questions btw: $$\int \frac{x^4\left ( 1-x \right )^4}{1+x^2} \, dx$$ $$\int \frac{x^4}{x^4+5x^2+4} \, dx$$ Thank you very much in advance!
uohzxela
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What does integrating a function wrt to another function intuitively mean?

I sometimes see stuff that is of the following notation: $$ \int f(x) dG(x) $$ I reckon a specific example would be: $$ \int x \operatorname{d \sin(x)} $$ How can one intuitively interpret this. Does this mean that we're compressing and stretching…
user1172468
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How do I calculate the integral of $\lfloor1/x\rfloor$ from $x=\frac{1}{n+2}$ to $x=\frac{1}{n}$?

How do I calculate integral : $$\int_{\frac{1}{n+2}}^{\frac{1}{n}}\lfloor1/x\rfloor dx$$ where $\lfloor t\rfloor$ means the integer part (I believe that's how it should be translated) or floor function of $t$.
Lola
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improper integral converge or diverge

I need to determine whether the improper integral converges or not: $\int_{0}^{\infty}\left(\frac{1}{2x}-\frac{1}{e^{x}-e^{-x}}\right)dx$ any ideas how to start? Edit: What I've done: $$\int_{0}^{\infty}\frac{1}{2x}-\frac{1}{e^{x}-e^{-x}} =…
user21312
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Integration by parts

I have done the 1st part of question and the answer I got is $\frac{5e^4 - 1}{32} $which I verified from the calculator too. But I am confused how to approach to the deducing part using previous result(since it is stated HENCE ). Any help is…
emil
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Unexpected outcome of integral

A given method for calculating $\int^1_{-1} \frac 1 {1+x^2} \, dx$ is \begin{align} & \int^1_{-1} \frac 1 {1+x^2} \, dx=\int^1_{-1} \frac 1 {x^2(1+\frac 1 {x^2})} \, dx = -\int^1_{-1} \frac 1 {1+(\frac 1 x)^2} \,d(1/x) = \left.-\arctan\left(\frac 1…
simp
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Integrate $x^ {1/2}e^{-x}$ using integration by parts

How to integrate $x^{1/2}e^{-x}$ using integration by parts? Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$
user2723
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Integration of $1/(1+a \csc^2(x))$

Integration of $$\int_{0}^{\frac{(M-1)\pi}{M} }\frac{1}{1+\alpha \csc^2(x)} dx,$$ where $ \alpha $ is a constant. I tried taking $\cot(x) = t$, then differentiating it w.r.t $dx$ we get, $-\csc^2(x)dx = dt$. And as we know that, $\csc^2(x)=…
Rahul Sharma
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Evaluate $\int_0^{2\pi} \frac{\cos 2x}{1-2a\cos x+a^2} $ with $a^2<1$

How do I compute the integration for $a^2<1$, $$\int_0^{2\pi} \dfrac{\cos 2x}{1-2a\cos x+a^2}dx=? $$ I think that: $$\cos2x =\dfrac{e^{i2x}+e^{-2ix}}{2}, \qquad\cos x =\dfrac{e^{ix}+e^{-ix}}{2}$$ But I cannot. Please help me.
Almot1960
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