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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integrate $\int \frac{dx}{x\sqrt{1-x}}$

$$\int \frac{dx}{x\sqrt{1-x}}$$ $$\int \frac{dx}{x\sqrt{1-x}}$$ $u=1-x$ $du=-dx$ $$-\int \frac{du}{(1-u)\sqrt{u}}$$ $a(1-u)+b\sqrt{u}=1\Rightarrow a-au+b\sqrt{u}=1$ $a=1\Rightarrow b\sqrt{u}-u=0\Rightarrow b=\sqrt{u}$ $$\int…
gbox
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integrate $\int \frac{dx}{4+3\sin2x}$

$$\int \frac{dx}{4+3\sin (2x)}$$ $u=2x$ $du=2dx$ $$\frac{1}{2}\int \frac{du}{4+3\sin(u)}$$ $v=\tan(\frac{u}{2})$ $du=\frac{2dv}{1+v^2}$ \begin{align*} \frac{1}{2}\int \frac{du}{4+3\sin(u)}={}&\frac{1}{2}\int…
gbox
  • 12,867
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Evaluate $\int_0^{\infty} \frac{x^2}{e^x-1} dx$

The problem is to show that the improper integral $I = \int_0^{\infty} \frac{x^2}{e^x-1} dx$ converges to $2\sum_1^{\infty} \frac1{n^3}$. Previously, I computed the following integral: $$f(x) = -\int_0^x\frac{\log(1-t)}{t}dt = \sum_1^{\infty}…
larryh
  • 207
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How Can I resolve this Integral?

I do not know if apply parts, or do a change: http://i.snag.gy/eeZam.jpg $\int x(2x+1)^{7/2}\, dx$
gabrielgbs97
  • 69
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Does the Lebesgue's criterion for Riemann-integrability also hold for Riemann-Stieltjes integral?

Lebesgue's criterion for Riemann-integrability: Let $f$ be defined and bounded on $[a,b]$ and let $D$ denote the set of discontinuities of $f$ in $[a,b]$.Then $f\in R$ on $[a,b]$ if,and only if,$D$ has measure zero. My question is,does the…
Luqing Ye
  • 780
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integrate $\int \cos^{4}x\sin^{4}xdx$

$$\int \cos^4x\sin^4xdx$$ How should I approach this? I know that $\sin^2x={1-\cos2x\over 2}$ and $\cos^2x={1+\cos2x\over 2}$
gbox
  • 12,867
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Integrate $∫\frac{\sec^2x}{(\sec x+\tan x)^{9/2}} dx$

How to integrate $$∫\frac{\sec^2x}{(\sec x+\tan x)^{9/2}} dx$$ In my book it is done like $$\begin{align} \sec x+\tan x &=t \\ \sec x−\tan x &= \frac 1t \\ (\sec x \tan x+\sec^2x)dx &=dt \\ \sec x(\sec x+\tan x)dx &=dt \\ \sec xdx &=\frac 1t dt…
user220382
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8 answers

solving $\int \cos^2x \sin2x dx$

$$\int \cos^2x \sin2x dx$$ $$\int \cos^2x \sin2x \, dx=\int \left(\frac{1}{2} +\frac{\cos2x}{2} \right) \sin2x \, dx$$ $u=\sin2x$ $du=2\cos2x\,dx$ $$\int \left(\frac{1}{2} +\frac{du}{4}\right)u \, du$$ Is the last step is ok?
gbox
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Solve $ \int {\frac{(x-1)dx}{(x-2)(x+1)^2 x^2 }} $

$$ \int {\frac{(x-1)dx}{(x-2)(x+1)^2 x^2 }} $$ I don't know about decomposition of fractions a lot,but I know that it is method which I have to use in this example. Please, help me.
K.Hurwitz
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Surface under the 'love formula'

To plot the love formula, you can plot two functions in one diagram: $$y_1(x)=\sqrt[3]{x^2}+\sqrt{1-x^2}+1$$ $$y_2(x)=\sqrt[3]{x^2}-\sqrt{1-x^2}+1$$ Then you get: Plot love function I calculated the surface inside the heart as, am I right of…
user301174
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Evaluation of $ \int_{0}^{\infty}\frac{\sin (kx)\cdot (\cos x)^k}{x}dx\;,$ where $k\in \mathbb{Z^{+}}$

Evaluation of $\displaystyle \int_{0}^{\infty}\frac{\sin (kx)\cdot (\cos x)^k}{x}dx\;,$ where $k\in \mathbb{Z^{+}}$ $\bf{My\; Try::}$ Let $$I(k) = \int_{0}^{\infty}\frac{\sin (kx)\cdot (\cos x)^k}{x}dx$$ Then $$I'(k) =…
juantheron
  • 53,015
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Integral of delta function and derivative of delta function

Can anyone rigorously prove this? $$\int dx \, \delta(x-\alpha)\delta^{\prime} (x-\beta) = \delta^{\prime} (\alpha-\beta).$$
Victor V Albert
  • 381
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Evaluating A Path Integral In Polar Coordinates

Show that the path integral of $f(x,y)$ along a path given in polar coordinates by $r=r(\theta)$ where $\theta_1 ≤ \theta ≤ \theta_2$, is $$\int_{\theta_1}^{\theta_2} f(r \cos \theta ,r \sin \theta) \sqrt {r^2+(\frac{dr}{d\theta})^2 } d\theta$$ I…
JAEMTO
  • 693
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Pulling out a constant in double integrals

$$ \iint {3x-y\over 9} \mathrm{d}x\mathrm{d}y$$ Is it safe to pull out a constant such as: $$ {1\over 9}\iint (3x-y) \ \mathrm{d}x\mathrm{d}y$$ I know this sounds silly, and it should be obvious that you can do this. But when I was trying to solve…
dendritic
  • 315
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Limitation of integration by parts to compute $\int e^{-x}(\cos wx + w\sin wx)dx$

I tried to solve the following question by integration by parts but it gets iterative and no solution is found. $$\int e^{-x}(\cos wx + w\sin wx)dx$$ where $w$ is constant. Is there any method by which this question could be solved? I don't want…
Sohail Ahmed
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