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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How to calculate this exponential integral ?

How to calculate this integral? : $$∫(1+a/x)^{x}dx$$ here $a∈ℝ^{∗}$.
DER
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Use the Shell Method to find the volume by revolving around the x-axis.

The functions are $x=\frac{y^2}{2}$, $x=2$, and $y=2$. I graphed it and it looks like the intersection points are $(2,2)$ and $(0,2)$. But I don't know how to set up the integral.
Pia
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Solving integral without fundamental theorem of calculus

Solve $\int_0^1 3xdx$ without using the fundamental theorem of calculus. I know that, to solve an integral without the fundamental theorem of calculus, I can find the upper sum and the lower sum. I can write down these sums. However, since they have…
kiwifruit
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Integration by Parts-Riemann-Stieltjes integral

How would I integrate the following? $\int_a^\infty (w-a)dF(w)$ for any fixed $a$, where $F(0)=0$ and $F(w)$ is strictly increasing and converges to $1$ as $w\to \infty$ . I've started by using the formula I've seen for integration by parts for…
Riemann-Stieltjes
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Bounds and the Fundamental Theorem of Calculus

Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous. Fix $a \in \mathbb{R}$ and define $$ F(x) := \int_a^x f(t) \, \mathrm{d}t. $$ Every version of the Fundamental theorem of calculus (FTC) I've seen tells us that $F$ is differentiable for $x \geq…
Amateur
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Can anyone solve the following integral analytically?

I'm trying to solve the following analytically: $P(u) = {1\over 2\pi} \int^{+\infty}_{-\infty} e^{i ut} \int^{+\infty}_{-\infty} e^{-x^2\over2} e^{-i \alpha t x} dx dt $ Where $i$ is the imaginary unit, $\alpha$ is a real parameter. $x$, $u$ and…
user2350366
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Integrate $\sin(2x)/(1 + \cos^2x)$ with $u$-substitution

Here's how I started: $$\int\frac{\sin{(2x)}}{1+\cos^2 x} dx = -2\int\frac{-\sin x\cos x}{1+\cos^2 x} dx = -2\int \frac{u}{1+u^2}du $$ I know the answer ends up being this from here: $$ -\ln\left(1+\cos^2 x\right) $$ But I don't understand where…
Joshua Oliphant
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Calculating $\int_{-1}^1 (2-|x|)dx$

I try to calculate the following integral but am a bit confused with the |x| $$\int_{-1}^1 (2-\left |x \right |)dx$$ The antiderivative should be: $\begin{cases} 2x-\frac{x^2}{2} & x \geq 0 \\ 2x + \frac{x^2}{2} & x \lt 0 \end{cases}$ Is that…
Chris
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Integration of infinite sum

I have a signal that I want to sample using delta functions. The signal is: $x(t) = W^2sinc^2(Wt)$ and after the sampling we will have the signal $z(t)$. We know the form of the signal in the frequency domain, and it is: $$Z(f) = W \sum \limits…
jasonsaras
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Asymptotic behavior of an integral

I am interested in the integral \begin{align*} \int_{\epsilon}^{\infty}dx_1\int_{\epsilon}^{\infty}dx_2\int_{\epsilon}^{\infty}dx_3\int_{\epsilon}^{\infty}dx_4\,\frac{1}{(x_1+x_3)(x_1+x_4)(x_2+x_3)(x_2+x_4)}e^{-x_1-x_2-x_3-x_4}. \end{align*} This…
Matthew Dodelson
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Integration $\int\left(\frac1{(x-1)^2(x+1)^3}\right)dx$

everyone. I am having problems to solve this integral: $$\int\left(\frac1{(x-1)^2*(x+1)^3}\right)dx$$ Any hints will be appreciated
dmsmar
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Perimeter Integrals

For $r = 1 + \cos \theta, 0 \le \theta < 2\pi$ in 2D polar space calculate the length of $P$, the perimeter by: $$ \int_P \sqrt{(dx)^2 + (dy)^2} \tag{1} $$ by showing: $$ (dx)^2 + (dy)^2 = (dr)^2 + (rd\theta)^2.\tag{2} $$ I am unsure where to go…
H SJ
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Help with integral $\int_{-\infty}^\infty e^{iαx-α^2 t} dα$

Solve by integration $$\int_{-\infty}^\infty e^{-iαx-α^2 t} dα$$ solve by integration , this integral is - infinity to infinity and exponent value
maha
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Help computing integral

I've been desperately trying to solve the following integral without much success. $$I(u)=\int_1^u \frac{e^{-x} (2 x-1)}{\sqrt{x~(A~e^{-x}+1)-B \sqrt{x}}}dx,$$ where $A,B\in \mathbb{R}$ are constants such that the integrand has no…
PML
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Integration of $\int \frac{dx}{a+f^2(x)}$

I want to solve a integral of the form: $$ \int \frac{dx}{a+f^2(x)} $$ in my particular case I got $$ \int \frac{dx}{5+\cos^2(x)} $$ in my case I followed this process: $$ \int \frac{dx}{5+\cos^2(x)} \\ let \ t = tg(\frac{x}{2}) => dx =…
Siscia
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