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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Integration of a function where the integral is equal to 0

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. Suppose that $\displaystyle\int_a^b x^nf(x) \, dx=0$ for all $n\in\{ 0,1,2, \ldots \}$. Prove that $f=0$. (Hint. Consider $\displaystyle\int_a^b f(x) \, f(x) \, dx$.) Hello, I have been…
greg
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how to integrate sqrt(sin(x)).without using elliptic integration .

How do you solve $$ \int\sqrt{\sin(t)}dt $$ using the substitution method.
user103816
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Help needed with difficult integral $\int_{-\pi}^{\pi}\cos(a \cos(x)) e^{b \cos(x'-x)} dx $

I have quite a tricky integral to be solved. I tried already several things, but I couldn't find the solution yet. To know that it's definitely not analytically solvable would help as well for not spending more time than necessary on it. The…
help_seeker
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Convergence of type $1/(kn+r)$ type series

Find a necessary and sufficient condition on $A$, $B$, and $C$ for which the series converges and find the sum in case of convergent. $$\sum_{n=0}^\infty \frac A{5n+1}+\frac B{5n+2}+\frac C{5n+3}$$ I found $A + B + C = 0$. Is that right?
DeepSea
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Dominated convergance of $\frac{1-e^{-xt^2}}{t^2}$

$$\begin{align}f(x, t)&=\frac{1-e^{-xt^2}}{t^2}\\ F(x)&=\int^\infty_0f(x,t)\ \mathrm{dt}\end{align}$$ I need to show that $F$ is continuous on $\Bbb R^+$. $F$ is defined everywhere and $f$ is continuous with respect to $x$, and now I need some…
Jack M
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Definite integrals: Evaluate the integral

Evaluate: $$ \int ^{1/2}_{1/4} \frac{dx}{ \sqrt{x-x^2}}dx$$ can u help me with this? What is meant by the dx in the numerator? EDIT: ANSWER AS GIVEN IN THE BOOK $$ \int ^{1/2}_{1/4} \frac{dx}{ \sqrt{x-x^2}}dx$$ $$=\int ^{1/2}_{1/4}…
chndn
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Integration of $\ln(\sin\frac{1}{x})$ between $\frac{2}{\pi}$ and $+\infty$

I would like to determine if the following converges, and if possible compute it: $$\int^\infty_{2/\pi}\ln\left(\sin{1\over x}\right)\,\mathrm dx$$ A bit of thought and a glance at a computer generated graph reveals that this is a horrific function.…
Jack M
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Convergence of the beta function $\beta(x,y)$

I have a question regarding the beta function $\beta(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt$ which has been asked here (Show that the Beta Function $\beta (x,y)$ Converges When $x \gt 0, \space y \gt 0$), and an answer supplied, however to my mind…
dandar
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i cant find this integral $\int_0^1 x^{-x} \mathrm{dx}$

$$\int_0^1 x^{-x} \mathrm{dx} $$
user96636
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Find the complete integral of $(p+q)(px+qy)=1$.

I am stuck on the following problem that says: Find the complete integral of $(p+q)(px+qy)=1$,where $p={ \partial z \over \partial x},q={ \partial z \over \partial y}$. My Attempt: The given equation is : $f(x,y,z,p,q)=(p+q)(px +qy)-1$.…
learner
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A monster integral

I have this homework question in a first course of calculus. The instruction ask to solve the integrals by substitute a=6 and b=6 My guess is to try to solve it by split the integral in 5 more simple ones. I ask my teacher for some leads, he…
Haizum Skallah
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Calculate $\lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{\sin(px)}{x}\,\mathrm dx.$

Suppose $f\colon[0,a]\to \mathbb{R}$ with $a>0$, is monotonically increasing on $[0,a]$ and $f(x) \geq 0$ for all $x\in [0,a]$. Calculate $$\lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{\sin(px)}{x}\,\mathrm dx.$$ My idea is as follows. Let $t=px$, then…
MathNoob
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Is there any relationship between these two equal integrals $\int_0^\infty\frac{\sin x}{x + x^2}dx=\int_0^\infty\frac{\pi-2\tan^{-1}x}{2 e^x}dx$

With some intermediate derivation results I found the two integrals are exactly the same. But why? \begin{align} I=\int_0^\infty\frac{\sin x}{x + x^2}\mathrm dx=\int_0^\infty\frac{\pi-2\tan^{-1}x}{2 e^x}\mathrm dx \end{align} Both are equal…
MathArt
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I am not sure if the integral $\int \cos(x^{\frac34})dx$ has a solution

My integral is $\int \cos(x^{\frac{3}{4}})dx$. I did the substitution $u=x^{\frac{3}{4}}$ then $\frac{4}{3}u^{\frac{1}{3}}du=dx$ and my new integral is $$ \frac{4}{3}\int \cos(u)u^{\frac{1}{3}}du$$. Now my intuition is to try with integration by…
weymar andres
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Trying to learn integrals, like $ \int_{0}^{1} \frac{x^3 - 1}{1 + x^2} \,dx $

I'm trying to learn how to calculate integrals on my own but I don't quite understand it. I found the following exercise on the internet but I don't even know where to start. I'm really lost. Can you explain it to me at a level that is quite "for…
fcoterroba
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