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 $\int_{0}^{\infty}\frac{2x}{x^8+2x^4+1}dx=\frac{\pi}{4}$

$$\int_{0}^{\infty}\frac{2x}{x^8+2x^4+1}dx=\frac{\pi}{4}$$ $u=x^4$ $\rightarrow$ $du=4x^3dx$ $x \rightarrow \infty$, $u\rightarrow \infty$ $x\rightarrow 0$, $u\rightarrow…
gymbvghjkgkjkhgfkl
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Evaluation of $\int_{0}^{1}\frac{\ln x}{x^2-x-1}dx$

Evaluation of $\displaystyle \int_{0}^{1}\frac{\ln x}{x^2-x-1}dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\infty}\frac{\ln…
juantheron
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Simple integration: $\int_{-1}^1 \frac{1}{x^2} dx$

Consider the integral $\int_{-1}^1 \frac{1}{x^2} dx$ then as a Riemann integral it diverges, also as a Lebesgue integral it is $\infty$. However, for a moment forget $0 \in [-1,1]$ and "integrate": $$\int_{-1}^1 \frac{1}{x^2} dx = \left.…
user16015
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Integration validity of $\int\frac{1}{\sqrt{a^2 + x^2}}\,dx$

I'm just wondering if the following integration is valid. \begin{array}{l} \int {\frac{1}{{\sqrt {{a^2} + {x^2}} }}} dx\\ {\rm{Let }}{u^2} = {a^2} + {x^2}\\ 2udu = 2xdx\\ \frac{{du}}{x} = \frac{{dx}}{u}\\ {\rm{Let }}\frac{{du}}{x} = \frac{{dx}}{u} =…
Mike
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How find this integral $I=\int_{-1}^{1}\frac{dx}{\sqrt{a^2+1-2ax}\sqrt{b^2+1-2bx}}$

Show this integral $$I=\int_{-1}^{1}\dfrac{dx}{\sqrt{a^2+1-2ax}\sqrt{b^2+1-2bx}}=\dfrac{1}{\sqrt{ab}}\ln{\dfrac{1+\sqrt{ab}}{1-\sqrt{ab}}}$$ where $0
math110
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Integral of $\frac{ \sqrt{\cos 2 x}}{\sin x}$

I am trying to solve the integral of $\frac{ \sqrt{\cos 2 x}}{\sin x}$. I converted this to $(\cot^2 x - 1)^{1/2}$ but after this I am stuck. I am not able to think of a suitable substitution. Any tips?
user34304
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How find this integral $\int\frac{\sin{x}}{\sqrt{2}+\sin{x}+\cos{x}}dx$

Find the integral $$\int\dfrac{\sin x}{\sqrt{2}+\sin x+\cos x} \, dx$$ My idea: since $$\sin x+\cos x=\sqrt{2}\sin(x+\dfrac{\pi}{4})$$ so $$\int\dfrac{\sin x}{\sqrt{2}+\sin x +\cos x} \, dx=\int\dfrac{\sin x}{\sqrt{2}(1+\sin (x+\dfrac{\pi}{4})} \,…
math110
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How would you evaluate $I:=\int_ {0}^{\infty} \frac {\cos(ax)} {(x^2 + b^2)^n} \ \mathrm{d}x$?

Any pointers on how should I start? $$I:=\int_ {0}^{\infty} \frac {\cos(ax)} {(x^2 + b^2)^n} \ \mathrm{d}x$$
paranoidhominid
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Evaluate $\int_0^\infty\frac {\sin^4(x)} {x^4} \operatorname dx$

How to evaluate the definite integral: $$ \int \limits_0^\infty\frac {\sin^4(x)} {x^4} \operatorname dx $$ Also provide the reference to various theorems to be to used to evaluate it! Thank you.
Abir Mukherjee
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Integrals involving $\operatorname{artanh}$ and $\log$

1. APOLOGY I had previously asked a similar question, but that question was closed due to negligence on my part (too little information, lack of respect for the respondent, etc.). I am very sorry about this. I felt that in the future I must take…
Kei Tojo
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A difficult integral: $\int_0^{\pi/2} (\log \sin x)^2 \ dx $

$$ \int_{0}^{\frac{\pi}{2}} (\log \sin x)^2 \, dx = \int_{0}^{\frac{\pi}{2}} (\log \cos x)^2 \, dx = \frac{\pi}{2}\left\{ \frac{\pi^2}{12} + (\log 2)^2 \right\} $$ I tried integration by parts, but I ended up with $$ \int_{0}^{\frac{\pi}{2}} (\log…
spjoes
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Integrating left to right versus right to left.

OK, I understand that when integration is done left to right with respect to x increasing left to right (dx is positive), that the answer is positive, and vice versa when integrating right to left. But, as a reality check, there is something that I…
user77970
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Double integration with Indicator function

The integral of interest is: $$Q = \int_{\mathbb{R}^2}\int_{\mathbb{R}^2} I\left(\frac{1}{2}\frac{x_2^2 - x_1^2 + y_2^2 - y_1^2}{x_2-x_1} \in [0,1]\right) \nonumber \\ \times I\left(2\arcsin\left(\frac{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}}{2\sqrt{y_1^2…
Oscar
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Evaluating $\int _0 ^1 x^k(\ln x)^m dx$ for integer $k$ and $m$

Find a formula for $\displaystyle \int _0 ^1 x^k(\ln x)^m dx$ that works for all positive integers $k$ and $m$. Use integration by parts $m$ times with $k$ fixed.
Sam
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Prove: $\int_{0}^{\pi}\frac{x\tan x}{\sec x+\cos x}dx = \frac{\pi^{2}}{4}$

How to prove following? $$\int_{0}^{\pi}\frac{x\tan x}{\sec x+\cos x}dx = \frac{\pi^{2}}{4}$$
kalpeshmpopat
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