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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Confusion about double integral and iterated integral

Suppose $f(x,y)$ is a function such that the double integral $$\int_{X\times Y} |f(x,y)|\,\mathrm dA$$ is infinite. Then $\displaystyle \int_X\int_Y f(x,y)\,\mathrm dy\mathrm dx$ and $\displaystyle \int_Y\int_X f(x,y)\,\mathrm dx\mathrm dy$ may…
michael_carbon
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Excuse me, I tried to solve $\int \frac{1}{4\cdot \sin \left(x\right)-5\cdot \cos \left(x\right)}dx$ but i can't get the right answer .. any help?

Excuse me, I tried to solve this problem but i can't get the right answer .. any help? $\int \frac{1}{4\cdot \sin \left(x\right)-5\cdot \cos \left(x\right)}dx$
Ahmed samy
  • 83
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Fourier transform integral with denominator $x^2+y^2$

I'm working on a problem, and stuck with the following integral $$\int_{-\infty}^\infty \dfrac{e^{-izx}}{x^2+y^2}dx$$ How can we compute this integral?
JJ Beck
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Closed form of integral

How does one show that, $\forall \ n \in \mathbb{N}$ $$\int \frac{\ln^n|\tan(x)|}{\sin(x)}dx = \frac{e^{i\pi n}(\ln|e^{ix}-1|-\ln|e^{ix}+1|)}{2^n}+C$$ Is this possible to do without using any complex analysis theory like the residue theorem or…
Parseval
  • 6,413
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Finding $ \int_{0}^{1} 2x\sin(\frac{1}{x}) - \cos(\frac{1}{x})\ dx. $

How do we solve the following integral ? $ \int_{0}^{1} 2x\sin(\frac{1}{x}) - \cos(\frac{1}{x})\ dx. $ I tried to proceed by integration by parts but got stuck.
bhavesh
  • 756
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Calculating $ \int \frac{1}{x} \sqrt{\frac{x+a}{x-a}}\mathrm dx $

I want to evaluate $ \int \frac{1}{x} \sqrt{\frac{x+a}{x-a}}\mathrm dx $. $ x=a\cosh(2t), \int \frac{1}{x} \sqrt{\frac{x+a}{x-a}}dx= \int \frac{2\tanh(2t)}{\tanh(t)}dt= \int \frac{4}{1+\tanh^2(t)}\mathrm dt $ $ u=\tanh(t), \int…
Chon
  • 6,002
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How to tackle the following integration problem

I am stuck on the following problem from an exercise in my analysis book: Show that $$\int_0^4 x \mathrm d(x-[x])=-2$$ where $[x]$ is the greatest integer not exceeding $x$. I think I have to partition the interval $[0,4]$ into some suitable…
learner
  • 6,726
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Integration of $|f(x)|$

I would like to learn how to use the sign function $\operatorname{sgn}(x)$ to integrate the absolute value of an arbitrary function. I know that integrating piecewise should give me the same result, but sometimes trigonometric functions can become…
Hautdesert
  • 1,606
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Evaluating a surface integral

Denote by $m$ the surface measure on the unit sphere $B$ in $\mathbb{C}^2$. Consider an unitary $2\times 2$ matrix $U$ and suppose we have a function $f\colon B\to \mathbb{C}$ which is $m$-integrable. Is it true that $\int_B f(x)\,\mbox{d}x = \int_B…
Jorg P.
  • 31
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How find this $I_{n}=\int_{-1}^{1}\arccos{\left(\sum_{k=1}^{n}(-1)^{k-1}x^{2k-1}\right)}dx$

Find this value $$I_{n}=\int_{-1}^{1}\arccos{\left(\sum_{k=1}^{n}(-1)^{k-1}x^{2k-1}\right)}dx=\pi?$$ My try:…
math110
  • 93,304
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How find this integral $\int_{0}^{x}\left(\frac{1}{2}-\{t\}\right)dt$

Find the value $$F(x)=\int_{0}^{x}\left(\dfrac{1}{2}-\{t\}\right)dt$$ where$\{x\}=x-[x]$ my try: since $$F(x)=\int_{0}^{x}\left(\dfrac{1}{2}-t+[t]\right)dt$$ when $$k-1\le t<=k,[t]=k-1,k\in Z$$ because $x$ is not integer,and maybe $x\to…
math110
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Evaluation of integral using laplace transform

How to evaluate the integral $\displaystyle\int_0^\infty\frac{e^{-at}-e^{-bt}}{t}dt$? As this is Laplace transform of $\frac{1}{t}$ at $s=a$, I tried with division by $t$ property, i.e. division by $t$ and integration from $s$ to infinity, but I am…
aditya
  • 31
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Prove that $\int_a^bf(x)dx+ \int_{f(a)}^{f(b)} f^{-1}(y)dy=bf(b)-af(a)$

Let f' be continuous and positive on [a,b]. Prove that: $\int_a^bf(x)dx+ \int_{f(a)}^{f(b)} f^{-1}(y)dy=bf(b)-af(a)$ I've got the later steps of this covered using integration by parts, and I know that I need to use substitution on the integral…
Jamie
  • 41
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Which integral limits to choose when dealing with open intervals?

Let's say I need to find out the length of an arc between the open interval $(a, b)$ with $a,b\in\mathbb{R}$. How would I set the limits for the integral? Am I still allowed to use $a$ and $b$? Adding (subtracting) some small $\epsilon$ to $a$…
HomerS1995
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$\int_{-\infty}^\infty \exp\{ -\frac{1}{2} (n+\frac{1}{k})(\mu-\frac{\frac{\varepsilon}{k}+\sum_{i=1}^n x_i}{n+\frac{1}{k}})^2 \} \; d\mu$

How do I integrate $$ \int_{-\infty}^\infty\exp\left(\vphantom{\Huge A^{A}}% -\frac{1}{2}\left[n + {1 \over k}\right] \left[\mu-\frac{\varepsilon/k + \sum_{i = 1}^{n}x_i}{n + 1/k}\right]^2 \right) \; d\mu$$ The answer is supposed to be…
Jiew Meng
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