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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Prove the uniform convergence involving logarithm

The function is like: $$F(x)=\int_{0}^{+\infty}(\ln t) (t^{x-1} )(e^{-t}) dt $$ Prove the integration's uniform convergence on $(0,+\infty)$ Well, you may notice that this is the derivative of $\Gamma$ function. The part I feel hard is how to…
Zhen Zhang
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Please check my solution of $\int \sin^6(x)\cos^3(x) dx$

$$\int \sin^6(x)\cos^3(x) dx = \int \sin^6(x)(1-\sin^2(x))\cos(x)dx$$ $$\int \sin^6(x)\cos(x)dx - \int\sin^8x\cos xdx$$ Now, $\cos xdx = d(\sin x)$ $$\int u^6du - \int u^8du = \frac{1}{7}u^7 - \frac{1}{9}u^9 + C$$ $$\frac{1}{7}\sin^7(x) -…
stil
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Integrate $ \cos^3x \sin^2 x \ \text{d}x$

$$\int \cos^3x \sin^2 x \ \text{d}x$$ I've simplified it to $\int \cos^3x-\cos^5x \,\mathrm{d}x$ using Pythagorean identities. It this the right way to do it? How do I integrate it from here?
Jim
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How to calculate the integral

$$\int_{0}^{+ \infty } \frac{\sin^{2} x}{x^{2}}dx$$ I suppose we should start from the known one:$$\int_{0}^{+ \infty } \frac{\sin x}{x}dx = \pi /2$$
Zhen Zhang
  • 805
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Limits of integration Confusion

Evaluate: $\displaystyle \int ^\infty _{-\infty} x^4e^{-x^2/2}dx$ If I notice this is an even function I can write this as : $2\displaystyle \int ^\infty _0 x^4e^{-x^2/2}dx$ If I then proceed with the substitution $u=\frac{x^2}{2}$ the limit of…
usainlightning
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Prove the integration is convergent

$$\int^{+\infty}_{0} \frac{ |\sin x| }{x(1+ \sqrt x)} dx$$ I try to set a upper boundary 1 for $| \sin x |$, but it doesn't work well This is initially to prove the absolute convergence
Zhen Zhang
  • 805
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A logarithmic integral question: $\int_{0}^{1} \frac{\ln^2{x}\ln^2{(1-x)}} {1+x}dx$

$$\int_{0}^{1} \frac{\ln^2{x}\ln^2{(1-x)}} {1+x}dx$$ Can the integral be calculated in terms of well known constants?
xuce
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Integrating $\int_0^1 \ln(1-t^{a}) dt . $

I would like to know how to calculate this integral $$ A= \int_0^1 \ln(1-t^{a}) dt . $$ I tried Taylor expansion for $\ln(1-t^{a})= -t^{a}$ , that gave me this : $$ A= \lim_{ x \rightarrow 0+} \int_0^1 -t^{a} dt =\dfrac{-1}{a+1} $$ is this result…
user2161721
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How to compute the integral of $\log x/(x+17)^2$ , x from 0 to $\infty$, using contour integration? Thanks!

How to compute the integral of $\log x/(x+17)^2$ , x from 0 to $\infty$, using contour integration? $$\int_{0}^\infty \frac{\log x}{(x+17)^2}dx$$ Thanks!
user145722
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Simplifying a Fourier integral

I have what is effectively a Fourier integral resulting from applying Abbe's theorem that I would like to simplify (ideally into a closed form solution): $$ f(\theta_0,\theta_1;\alpha) = \int_{\theta_0}^{\theta_1} e^{i \alpha \cos \theta} \cos…
Victor Liu
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Simple integral $\displaystyle\int \frac{e^x}{x^2-a^2}\ dx$

Is this integral solvable? $$\int \frac{e^x}{x^2-a^2}dx,\quad a>0.$$
David
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Definition of Surface and Line integrals

I am trying to understand the concepts of surface and line integrals better, and since I'm not a mathematician I have only come across explanaitons which are a bit over my head. I have a few fairly basic questions. 1. Can you give me simple…
turnip
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Why is the integral of $0.5/x = 1/2 \ln x$ and not $1/2 \ln 2x$?

When you calculate $\int\frac{1}{2x}dx$ you get $\frac{1}{2}\ln(x)$ and when you calculate $\int\frac{1}{2x}dx$ you don't get $\frac{1}{2}\ln(2x)$. $\frac{1}{2x}$ is the same as $\frac{1}{2x}$ why do you get different answers? …
Yusaf
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doubt in the proof of a theorem in baby rudin

Theorem: Suppose $f$ is bounded on [a, b],$f$ has only finitely many points of discontinuities on [a, b], and $\alpha$ is continuous at every point at which $f$ is discontinuous. Then $f\in R(\alpha)$. To prove this, we cover E, the set of points of…
matthew
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Integration with exponential constant

How can I find $$\int_0^\infty \frac{2\left(e^{-t^2} -e^{-t}\right)}{t}\ dt$$ I have been told the answer is $\gamma$, the Euler-Mascheroni constant, but do not understand how this is derived.
user71408
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