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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Simplifying $I = \int_{-\pi/2}^{+\pi/2} d\phi \left[ 1-M(\phi) \theta^*(\phi) \right]$

I am stuck at simplifying the following integral: $$ I = \int_{-\pi/2}^{+\pi/2} d\phi \left[ 1-M(\phi) \theta^*(\phi) \right] $$ where $ M= p \cos \phi + q \sin \phi $. Here $p,q$ are fixed. The $\theta^\star$ is defined through $\cos…
Steph
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Riemann zeta function and Bernoulli numbers

According to Wikipedia (https://en.wikipedia.org/wiki/Bernoulli_number) we have: $$1^m + 2^m + ... n^m = S_m(n) = \frac{1}{m+1} \sum_{k=0}^{m} \binom{m+1}{k} B_k^{+} n^{m+1-k}$$ where $B_k^{+}$ are the (positive) Bernoulli numbers. From plugging in…
Nikolai riber skånstrøm
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Solve for a variable inside a complicated integrand

I would like to solve the following equation for $b$. As it is now, Wolfram Alpha is not giving an answer. $$\frac{\pi\cot B}{A}=\int^{\pi}_0\left(\frac{\frac{b\cos…
rdemo
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A Radical Integral $\int\frac{x^2-3x+1}{\sqrt{1-x^2}}dx$

Evaluate: $$\int\frac{x^2-3x+1}{\sqrt{1-x^2}}dx$$ Please tell me where I committed the mistake as the answer I got does not match with the actual answer of $$=\frac{3sin^{-1}(x)}{2}+3\sqrt{1-x^2}-\frac{(\sqrt{1-x^2})x}{2}+c$$ Let…
Crustocean 01
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Evaluate $ \int \cfrac { \sec^2x}{ ( \sec x + \tan x)^{9/2}}dx $

Problem : Evaluate $ \int \cfrac { \sec^2x}{ ( \sec x + \tan x)^{9/2}}dx $ Solution : $ \int \cfrac { \sec^2x}{ ( \sec x + \tan x)^{9/2}}dx $ $ \int \cfrac { \sec^2x}{ \left(1+ \sin x \over \cos x \right)^ {9/2}}dx $ $ \int \cfrac { \sec^2x}{…
rst
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Quantic Physics Integral

I encountered this integral in my Quantum Mechanics class archives (which was apparently asked during an oral exam in the past years) : $t \in \mathbb{R}$ is a parameter $$I(t) = \int^{\frac{1}{2}}_{-\frac{1}{2}} \sin(t(x^2-x^4))dx$$ To be clear…
gazo
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Let $M:=\left\{(x, y) \in \mathbb{R}^2: x^2+y^2 \leq 1\text{ and }(x \leq 0\text{ or }|y| \leq x)\right\}$. Determine $\lambda_2(M)$.

I think I need Cavalieri: $(X, \mathcal{A}, \mu),(Y, \mathcal{B}, \nu)$ are $\sigma$-finite measure spaces. For $E \in \mathcal{A} \otimes \mathcal{B}$ is $$ \mu \otimes \nu(E)=\int_X \nu\left(E_x\right) \mathrm{d} \mu(x)=\int_Y \mu\left(E^y\right)…
MathJason
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Two STEP level integration questions

I can't see what to do in the two STEP level questions below. Can anyone give me a clue, please? The function $f$ satisfies the condition $f'(x) > 0$ for $a \leq x \leq b$, and $g$ is the inverse of $f$. By making a suitable change of variable,…
gary
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Solving a recursive integral

I'm trying to compute the following integral: $$ \lambda m(x) =\frac{1}{2\varepsilon} \int_{\max(\alpha x - \varepsilon, -1)} ^{\min(\alpha x + \varepsilon , 1)}m(y) \mathrm{d} y, $$ with $\alpha$ and $\varepsilon$ positive constants. Assuming…
BB3C
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Evaluating $\int\frac{dr}{r\sqrt{ar^2+br+c}}$, Landau's 'elementary' integral

This has probably been asked countless times before, but is there a particularly elegant method of evaluating$$\displaystyle I=\int\frac{dr}{r\sqrt{ar^2+br+c}}$$ ,with $a,b>0,c<0$? Background: The integral occurs when calculating the angle, $\phi$,…
Meow
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Please evaluate the following integral with the respective constraints.

Let $x^2+1 \neq n \pi$ and $2 x^2+1 \neq n \pi \,\forall n \,\in \mathbb{N}$, then $$ \int x\sqrt{\left|\frac{2\sin(x^2+1)-\sin(2x^2+1)}{2\sin(x^2+1)+\sin(2x^2+1)}\right|} dx=? $$ This problem is inspired from a really easy problem from JEE mains…
Aditya Naskar
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How to evaluate $\int_0^{\infty} \frac{\cos{x}}{1+x^4} dx$ without contour integration?

Using contour integration I found that $\int_0^{\infty} \frac{\cos{m x}}{1+x^4} dx = \frac{\pi}{2\sqrt{2}} e^{-\frac{|m|}{\sqrt{2}}} (\cos\frac{m}{\sqrt{2}} + \sin\frac{|m|}{\sqrt{2}})$ Is there any way to evaluate the integral using the…
Archisman Panigrahi
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Can one say "take an integral" instead of the usual "calculate an integral"? What other options are there?

Russians often use the formulation "take an integral". Now I noticed it in an article, written by a Russian speaker and I can't recall I have ever encountered it in English. Is it a possible alternative to the basic "calculate an integral"? What…
wondering
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Gaussian integral method for double integral

How to solve a double integral using Gaussian integral method? $$\iint _\Omega \frac{xy}{1 + x^2 + y^2}\,\mathrm{d}x\,\mathrm{d}y$$ Where $\Omega$: $0\le x\le 1$ and $0\le y\le 1$.
Jonathan
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Integration by Substitution solved.

I want to solve this integral. For integration by substitution, I let $$\frac{d}{dt}f(t)=\tan\theta.$$ But, I cannot solve for $f(t)$ and $\frac{d^2}{dt^2}f(t)$. I cannot even find $\frac{dt}{d\theta}$. Can anyone help me?
qmoi
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