Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.
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Need help plugging in bounds in a u-substitution
This is a really basic question but I'm just a little confused. I have this integral: $$\int_{0}^{2\pi}\cos^3tdt$$ I solved it by doing a u-sub: $$\int_{0}^{2\pi}\cos t(1-\sin^2t)dt$$ Let $u=\sin t$, ... calculus integration definite-integrals- 598
Evan's statement about integral of the divergence of a vector field.
At the start of Evan;s textbook there is this statement: How can we conclude 3 from the first equality? Why can't it be that the divergence is positive and negative over the volume in equal ... calculus integration partial-differential-equations divergence-theorem- 2,301
Calculating the integeral $\int_0^{\infty}\frac{{(e^{-ax}-e^{-bx})}{\cos(cx)}}{x}dx$
I want to calculate:$$\int_0^{\infty}\frac{{(e^{-ax}-e^{-bx})}{\cos(cx)}}{x}dx(a,b,c > 0)\tag{1}$$And I want to use:$$\frac{x}{x^2+k^2}=\int_{0}^{\infty}e^{-xy}\cos{ky}dy\tag{2}$$So,I want to think:... real-analysis integration analysis fubini-tonelli-theorems- 11
Is there any method other than Feynman’s Integration Technique to find $ \int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x?$
We are going to find the formula, by Feynman’s Integration Technique, for $$\int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x,$$ where $a+c$ $\textrm{ and }$ $b+c$ are ... calculus integration definite-integrals- 5,342
Asymptotics of a two dimensional integral
I am working on the following integral $\int_0^1d\epsilon\int_{-\epsilon}^\epsilon dt (\sqrt{1-(\rho+t)^2}-\sqrt{1-\rho^2})e^{-N t^2},$ where $\rho=1-\epsilon$, $N\rightarrow \infty$. The problem is ... integration definite-integrals asymptotics approximate-integration- 73
Non-obvious Trigometric Integral
Prove that for every $\alpha\ge 0$ then $$\int_0^{\pi/4} (\sin(t)\cos(t))^\alpha(1-(\alpha+2)\cos^2(2t)) = 0$$ This is so non-obvious to me. integration trigonometric-integrals- 2,388
Integral over perpendicular coordinates
I have to calculate the following integral: $$ S[z] = \int \frac{d^d x d^d y}{|x_\perp|^{d-\alpha} |y_\perp|^{d-\alpha} |x-y|^{2\alpha-\beta} |z-x|^\beta |z-y|^\beta}, $$ where $x$ belongs to a $d$-... integration special-functions conformal-field-theory- 1
Calculation of this integral
$z$ is a positive real number $$F(z)=\int\limits_{0}^{2\pi} \dfrac{\cos^4(x) \ln(1+z^2 \cos^2(x))}{1+z^2 \cos^2(x)} \, dx$$ In the aim to simplify the calculation , i first make a derivative of $F(z)$ ... real-analysis integration- 1
Analytical solution of an integral involving gaussian
I was wondering if there is any analytical solution to the following integral: $$\int_L^U\frac{e^{-\frac{(x-a)^2}{2\sigma^2}}}{x} dx$$ with $\sigma, L, U>0$. integration definite-integrals- 514
Integral with weierstrauss substituion, limits gone wrong
I'm trying to evaluate the following integral with a Weierstrass substitution: $$ \int_{\pi/3}^{4\pi/3} \frac{3}{13 + 6\sin x - 5\cos x} \text{d}x $$ This comes out to about 0.55 when evaluated ... integration- 81
Problem with the integral $\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t$
I'd like to prove, using a partial fraction decomposition (I don't want to use residue calculus), that $$\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t=\frac{\pi}{2n\sin\frac{m\pi}{2n}}$$ where $1\... integration improper-integrals- 4,644
Integral result contains one term is infinite large and one term is infinite small.
I am trying to integrate this function from $0$ to $f$. $ \int_0^f\dfrac{p\cdot\left(\mathrm{e}^{-\frac{\ln\left(r\right)\,x}{\left(r-1\right)t}}-\mathrm{e}^{-\frac{r\ln\left(r\right)\,x}{\left(r-1\... calculus integration definite-integrals- 21
Understanding the proof that $f\in L^1_{loc} =0 $ a.e if its integral with all test function is 0
I know this question has been asked before but there is one specific point about this proof that is bothering me : So I was wondering what was the point of extending $f$ to the whole on $R^n$ until I ... integration analysis calculus-of-variations- 3,103
Asymptotics of an integral with singular derivation
I want to evalute the leading order term of the following integral as a series of $1/N$ and $\epsilon$, $\int_{-\epsilon}^\epsilon dt (\sqrt{1-(\rho+t)^2}-\sqrt{1-\rho^2})e^{-N t^2}$, where $\rho=1-\... integration definite-integrals asymptotics approximation- 73
Convolution Equation Solving
Solving the equation $$y'(t)=\sin(5t)-25 \int\limits_{0}^{t} y(u)\,\mathrm du$$ with $y(0)=0$, we obtain the convolution $$y(t)=\sin(5t)\cdot g(t)$$ For some function $g(t)$. What is the value of $g\... integration ordinary-differential-equations laplace-transform convolution- 91
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