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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Does this integral have a closed form solution?

Does anyone know of a method by which to tackle the following rather nasty surface integral? $$ \iint_{\theta\in[0,\pi], \phi\in[0,2\pi)} \left[\sigma^2[\left(x-y+R\sin{\theta} \right)\left( \cos{\phi}-\sin{\phi}\right)]^2 +[y+\left(z-\rho+R…
JoshD
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Integral of gravitational acceleration

You probably know the the pull of gravity varies with the inverse square of the distance $(r = x)$ $x^2$. We all know in physics that the integral of $\frac{K}{x^2}$ is $-\frac{K}{x}$, this when the attracted body is free-falling. I'd like to ask…
user157860
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Function with a discontinuity in its first derivative

I want to compute this integral, $I = \int_{-\infty}^{+\infty} \psi(x)\frac{d^2}{dx^2} \psi(x) dx$. Now, consider a function which has a discontinuity in its first derivative. For an example I'll consider a triangular wave centered at $\frac \Delta…
sbp
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Integral of $\cos(x)\exp(-x^2/2\sigma^2)$

I have to evaluate $$ with $x$ distributed according to $Q^{-1}e^-{\frac{x^2}{2\sigma^2}}$. I have gone this far: $$=\frac{\int\limits_{-\infty}^\infty \cos (x) Q^{-1}e^{-\frac{x^2}{2\sigma^2}}…
Džuris
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Integral involving 2-dimensional Gaussian function

I'm trying to evaluate the following integral (or at least get a bound involving $s_x$ and $s_y$): $$ \int_0^\infty\int_0^\infty \frac{xy}{(x^2+y^2)^{3/2}}\exp\left\{-\frac{1}{2}\left(\frac{x^2}{s_x^2}+\frac{y^2}{s_y^2}\right)\right\}dy…
Robert W.
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What is the solution of this definite integral: $\int_{0}^{2\pi} \ln\big(C-D\cos(f)\big)\cos(nf)df$

I encountered this integral in my calculations: $$\int_{0}^{2\pi} \ln\big(C-D\cos(f)\big)\cos(nf)df$$ Here, $n$ is a natural number, $C, D$ are constants, such that $C\gt D$. I tried to find solution in Table of integrals,series and products and by…
t387
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Can this cylindrical shell integral problem be done with washers instead?

Instead of using shells, can I do this problem via horizontal integration and subtracting the smaller circle from the larger circle and use washers instead? Is the volume just: $\int$ Area of larger circle - Area of smaller circle \, dy ? outer…
Jwan622
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Mathematica can't solve this integral, any tips?

$$0.5 \int_{-\infty }^{\infty } \frac{e^{-\frac{x^2}{4 (c+k L)^2}} -\frac{-2 c e^{-\frac{x^2}{4 c^2}} +\frac{\sqrt{\pi } x (\text{erf} x)}{2 (c+k L)} -\frac{\sqrt{\pi } x (\text{erf} x)}{2 c} …
Craggles
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Value of $k$ in ratio of two definite integration

If $\displaystyle I_{1}=\int^{1}_{0}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{12}dx$ and $\displaystyle I_{2}=\int^{1}_{0}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{(3+x)^8}dx$ and $I_{1}=k(108\sqrt{3})I_{2}$. Then $k$ is Try: put…
DXT
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Complex definite integral involving Trigonometric function

Evaluation of $$\int^{\frac{\pi}{2}}_{0}\frac{1}{\sqrt{1+\sqrt{\tan x}}}dx$$ Try: Put $\tan x=t^2$ and $\sec^2 x dx=2tdt$. $$I=\int^{\infty}_{0}\frac{2t}{(1+t^4)\sqrt{1+t}}dt$$ Put $1+t=u^2$ and…
DXT
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Finding $\int^{\pi}_{0}\frac{1}{\sqrt{a-\cos x}}dx$

Finding $$\int^{\pi}_{0}\frac{1}{\sqrt{a-\cos x}}dx$$ Try: using $$\cos x =\frac{1-\tan^2\frac{x}{2}}{1+\tan^2\frac{x}{2}}$$ $$\int^{\pi}_{0}\frac{\sec \frac{x}{2}}{\sqrt{(a-1)+(a+1)\tan^2\frac{x}{2}}}dx$$ Could some help me to solve it , thanks
DXT
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Rational Gaussian-type integral with sin

Here is a tough integral. Does anyone have any clever ideas?. I have tried all sorts of things, but make no real headway. $\displaystyle \int_{0}^{\infty}\frac{e^{-x^{2}}\sin^{2}(x)}{x^{2}}dx$ Would residues be a consideration with this one?.…
Cody
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Integral of sin(x)

I often hear that $$\int_0^x \sin(x)\,dx$$ is equal to $\cos(0)−\cos(x)$: Now if we take the limit as $n \to \infty $ we see $ \frac{x}{2n} \to 0$ and $$\int_0^x \sin t \, dt = \lim_{n \to \infty}\frac{x}{n}\sum_{k=1}^n\sin \left(\frac{kx}{n}…
Shreyas Shridharan
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$\int_0^{2\pi} \left(a\cos x+\sqrt{1-a^2\sin^2(x)}\right)^{2k} dt=2\pi\sum_{m=1}^k \binom{k}{m} \binom{k-1}{m-1}a^{2(k-m)}$

I would like to know if there is a higher-level perspective to explain this result (or a simple way to prove it)? Thanks for help. $$\int_0^{2\pi} \left(a\cos x+\sqrt{1-a^2\sin^2(x)}\right)^{2k} dt=2\pi\sum_{m=1}^k \binom{k}{m}…
ZENG
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definite integral with arctan of length-like quantity

I am trying to show that for $0
Reinstate Monica
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