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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integral on the surface of a sphere

How can I calculate the integral of $f(z) = e^{-z}$ over the surface of a sphere with radius $R$? I tried using cylindrical and spherical systems, both gave an unsolvable integral, suspecting there's a way to change the order of the variables.
Jonathan
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Proof that improper integration works

I wasn't really sure how to title this, but the way the problem is phrased makes me think of improper integrals so I thought that might be a good title. Let $f\colon[a,b]\to\mathbb{R}, a
chris
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Help to solve: $\int 1/(x\sqrt{25-x^2})\ dx$

I'm a brand new student. Need some help to integrate this. Perform this integration: $$\int \frac{1}{x\sqrt{25-x^2}}\ dx$$ I'm able to obtain in theta terms like this: $$\frac 15 \ln⁡|\cscθ-\cotθ|+C$$ But I have problems to convert in terms of…
El_Master
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How to evaluate $\int \frac{\sin (\pi x)}{|x|^a + 1} dx$

How do I integrate this? $$\int \frac{\sin (\pi x)}{|x|^a+1} dx$$ I really struggle to find a solution. I even tried Wolfram Alpha and Mathematica, but neither could give me an answer. I have to find all $a$ for that the Improper Integral exists.…
krnflake
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Is it possible to integrate a greatest integer function?

Is there an indefinite integral for this function ? $$\int [x] dx$$ I know how to integrate it if it was something like this $$\int_b ^a [x] dx$$
DocDev
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Integrate $\int \cos^n x dx$

I know how to solve it with reduction formula, but is there concrete answer for this integral (without other integrals like in reduction formula)? WolframAlpha gives me expression with $F_1(\frac{1}{2}, \frac{n+1}{2};\frac{n+3}{2}; \cos^2(x))$, but…
Shmuser
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Help! Integrate using substitution method.

I need help integrating the following function: $$\int\frac{2x+5}{\sqrt{16-6x-x^2}}dx$$
Andres Salazar
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A parametric Gaussian-like integral

$$ \int\limits_0^{+\infty} e^{-a^2x^2-\frac{b^2}{x^2}} dx$$ I'm stuck after doing this: $$ I = e^{-2ab}\int\limits_0^{+\infty} e^{-(ax-\frac{b}{x})^2} dx$$
Mannix
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Problem with integrating sine function

I have a problem solving an integration. This is my approach: \begin{align*} u &= \frac{1}{T}\cdot\intop_{0}^{T}(U_{0}+\hat{u}\cdot \sin(\omega t))dt\qquad\text{with }\omega=\frac{2\pi}{T}\\ u &= \frac{1}{T}\cdot\intop_{0}^{T}(U_{0}+\hat{u}\cdot…
svenwltr
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Calculate :$\int_{0}^{\infty}{ }\frac{dx}{a^6+(x-\frac{1}{x})^6}$

Find the: $$\int_{0}^{\infty}{ }\frac{dx}{a^6+(x-\frac{1}{x})^6}:a>0$$ My Try: $$\frac{1}{a^6}\int_{0}^{\infty}{…
Almot1960
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Calculate integral of fractional part

I have to calculate $$I=\int _0^1\left\{nx\right\}^2dx , \:\:\:\:\: n \in \mathbb N, n \ge 1$$ Where {a} is $frac(a)$. I know that $\left\{nx\right\}^2 = (nx - [nx])^2$ so $$I\:=\int…
Liviu
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Logarithmic divergence of an integral

I would like to prove that the following integral is logarithmically divergent. $$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-x)^{2}-a)((k-y)^{2}-a)((k-z)^{2}-a)}$$ This is 'obvious' because the power of $k$ in the numerator is $4$, but the highest power…
nightmarish
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Antiderivative of $\sin(\frac{1}{x})$

I have to find the values of parameter $a$ so the function $ f:\mathbb R \to \mathbb R, f(x) = \begin{cases} \sin\frac{1}{x} & \text{ for } x \in \mathbb {R}\setminus\{0\}\\ a & \text{ for } x = 0 .\end{cases}$ has an antiderivative on $\mathbb R$.…
Liviu
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Closed form for the integral $\int_0^\infty t^s/(1+t^2)$

I want to find a following integral $$\int_0^\infty \frac{t^s}{1+t^2} \,\frac{dt}{t}$$ where $s \in \mathbb{C}$, $\Re(s) \in (0, 2)$ and want to find a closed form for it. I think it should be $\frac{1}{2} \Gamma(s/2)\Gamma(1-s/2)$ but I'm not…
MT_
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$\int_{-1}^{1}e^{-\frac{1}{1-x^2}}dx$, can it be computed?

Is there a way to compute $\int^{1}_{-1} e^{-\frac{1}{1-x^2}}dx$ ? I have tried a few change of variables and also to write down $\frac{1}{1-x^2} = \frac{1}{2} ( \frac{1}{1-x} + \frac{1}{1+x})$ But I didn't get anything so far. Edit: changed…
incas
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