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.

73636 questions
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What does one mean when $\int \frac{\sin x}x$ doesn't exist?

Well I say that by taylor's expansion: $$\int\frac{\sin x}x=\int\frac{x-x^3/6+x^5/120+...}x=x-x^3/18+x^5/480+...+\mathbb{C}$$ It's another thing that there doesn't exists a closed form for the sum/difference.But it does exists.So I am now confused…
RE60K
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Primitive of $\frac{3x^4-1}{(x^4+x+1)^2}$

How to find primitive of: $$\frac{3x^4-1}{(x^4+x+1)^2}$$ I am having a faint idea of a type which may or maynot be in the primitve, i.e.: $$\frac{p(x)}{x^4+x+1}$$ The problem is I am not getting an idea of a substitution to solve this problem. I…
RE60K
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Indefinite integral of $\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}$

How do I find $$\int\frac{2x^3 + 5x^2 +2x +2}{(x^2 +2x + 2)(x^2 + 2x - 2)}\mathrm dx$$ I used partial fractions by breaking up $x^2 + 2x - 2$ into $(x+1)^2 - 3$ and split it into $(a+b)(a-b)$ but as u can see it's extreme tedious. I was wondering if…
Danxe
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Evaluate $\int\frac{\sqrt{9-x^2}}{x^2}\mathrm dx$

I am trying to solve $$\int\frac{\sqrt{9-x^2}}{x^2}\mathrm dx$$ My answer is slightly different to the memo: $x=3\sin\theta\quad\iff\quad\theta=\arcsin\left(\frac x 3\right)\\ \text dx=3\cos\theta\ \text…
ahorn
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How to prove this double integral identity with simaler Jenson decomposition?

Question: let $x=(x_{1},x_{2},\cdots,x_{n})$,and $f:I\to R$have two derivative,and for any $[a,b]\subseteq I$,then $f''$ is integrable on $[a,b]$,and $x\in I^n,n\ge 2$ show that $$J[f(x)]=\dfrac{1}{n^2}\sum_{1\le i
math110
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Showing equivalence of the hyperbolic function

I've been thinking about this problem for about 30 minutes now and still I don't have a clue on what I'd do next. I have this equation and I aim to show equivalence (proof)…
wildberry tart
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Chain rule in integration?

Why is it that integration of 1/(xlogx) is log(log x) and not log(log x) (the integration of log x)? See when we do the u substitution we get the answer but doesn't chain rule apply for integration? For eg- integration of (3(x)^2)/(1+(x)^2) We do u…
geek101
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$\int \cos^{-1} x \; dx$; trying to salvage an unsuccessful attempt

$$ \begin{align} \int \cos^{-1} x \; dx &= \int \cos^{-1} x \times 1 \; dx \end{align} $$ Then, setting $$\begin{array}{l l} u=\cos^{-1} x & v=x \\ u' = -\frac{1}{\sqrt{1-x^2}} & v'=1\\ \end{array}$$ Then by the IBP technique, we…
ptrcao
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Finding the volume of $(x-4)^2+y^2 \leqslant 4$

Calculate the volume of the ring-formed body you get when the circle formed area $$(x-4)^2+y^2 \leqslant 4$$ rotates around the y-axel. The answer should be: $32\pi^2$ My approach was: $$ \pi \int_2^6 \left(\sqrt{(x-4)^2-4}\right)^2 dx $$ but I…
iveqy
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Integration with two unknowns

I'm completely stumped with this one, I'm not sure how I should do this. The equation of a parabola is $y=-3x(x-2)$. It intersects the $x$-axis at $0$ and $2$. Given that the area of this parabola is $4\,{\rm units}^2$, there will be a straight line…
user133422
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integrating square root of tanx

$\int \sqrt{\tan (x)}dx $ Let $\tan(x)=t^{2}$ then $dx$ will become $\frac{2t}{1+t^{4}}$ Hence $\int \sqrt{\tan (x)}dx =\int\frac{2t}{1+t^4} dt $ But I cannot proceed from this step.
Madhu
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Integration - finding an explicit formula

The question in my textbook asks: If $f$ is a continuous function such that $$\int\limits_0^x{f(t)dt}=xe^{2x}+\int\limits_0^x{e^{-t}f(t)dt}$$ for all $x$, find an explicit formula for $f(x)$. My working goes as follows: I decided to analyse the…
ahorn
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Proving the indefinite integral $ \int\frac{\sqrt{a+bu}}{u}du $

How can I prove that the indefinite integral $ \int\frac{\sqrt{a+bu}}{u}du $ is equal to $ 2\sqrt{a+bu} +a \int\frac{1}{u\sqrt{a+bu}}du$
user8028
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How do textbooks display the mean by integration

My question may be expressed with an example I'm most familiar with.. In many ocean science text books, the mean density of the water column is expressed as $${\widehat\rho}=\int^0_{-h}\rho(z)dz$$ How can this be the mean? Surely you need $1/(b-a)$…
Mark
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Problem with volume integral of a scalar function

I have difficulties in integrating this scalar function over the assigned volume. Let $D=\{(x,y,z):(x-2z)^2+(y-x)^2+(x+z)^2\le4,\,0\le x+y+z\le1\}$ Calculate $\int_D z\,dxdydz$
Fuzzee Lowgeek
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