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.

73636 questions
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Why $\int_{0}^{\pi/2}\tan(x/2) dx= \ln 2$

I don't know why $$\int_0^{\pi/2}\tan\frac{x}2\ dx= \ln 2.$$ How can i solve this to get that answer?
Soroush
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I don't know why $\int_{-3}^{-2} \frac{dx}{x} = \ln(\frac{2}{3})$

I don't know why $$\int_{-3}^{-2} \frac{dx} x = \ln \frac{2}{3}.$$ How can i solve this to get that answer?
Soroush
  • 342
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Integration of $|f(x)|$ or $\sqrt{f(x)^2}$ or $\left(\sqrt{f(x)}\right)^2$ as $f(x)$ tends towards chaotic

Assuming $f(x)$ is a Real function of a Real variable for each "$\sqrt{f(x)^2}$", then as $f(x)$ tends from no crossings of $f(x) = 0$ to chaotic & dense crossings of $f(x) = 0$, does computation of a definite integral remain simple or does it…
Valentine
  • 31
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Find $\int \frac{1}{\sin\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}dx$

$\int \frac{1}{\sin\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}dx$ I tried to solve it.I put $\frac{x}{2}=t$ then $\int \frac{1}{\sin\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}dx=$ $\int \frac{2 dt}{\sin t\sqrt{\cos^3 t} }=\int \frac{2 dt}{\sin t \cos t\sqrt{\cos…
diya
  • 3,589
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calculate $\int_o^\pi \frac{x\sin x}{1+\cos^2x} \,dx$

Given $$I = \int\limits_0^\pi \frac{x\sin x}{1+\cos^2\!x}dx$$ prove without calculating $ \frac{\pi}{2} \le I \le \pi $ calulate $I$ so far I know $I = \int\limits_0^\pi \frac{x\sin x}{1+\cos^2\!x}dx \le \int\limits_0^\pi x\sin x =\pi$
Daniel Katzan
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Integration by parts with surface integrals

I am working with some kinetic theory. I have the distribution function $\Psi (\overrightarrow{r},\overrightarrow{p},t)$, Where $\overrightarrow{r}$ - radius vector, $\overrightarrow{p}$ - unit vector of orientation, $t$ - time. But my question is…
Mikhail Genkin
  • 194
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$\int_{a}^{b}\left(\frac{\cos(x)\tan^{\pi}(x)}{\sin^3(x)}\right)dx$

I've got an integration problem; I don't know how to go from the left 'red' side to the right. Can someone help me? Assuming $a
Jan Eerland
  • 28,671
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How to find: $\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$?

How to find: $$\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$$ I tried multiplying it all out but I just ended up in a real mess and I'm wondering if there is something I'm missing.
Eldat P
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Evaluate the integral $\int \frac{x}{a+bx^3}\ dx$

How do I solve integral at this form $\displaystyle\int \frac{x}{a+bx^3}\ dx$ ?
Hamid Mohammad
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How to integrate $\int \frac{\arctan x}{x^4} dx$?

I have written the integral as $\int x^{-4} \arctan x dx$. Then, by applying by parts, I got $-3\dfrac{\arctan x}{x^3} + 3\int \dfrac{1}{x^3(1 + x^2)} dx$. Now, how can I solve the later integral? Is there any other trick to do this?
user142971
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Integrate $\int{2^{2x}} dx$

Integrate $$\int{2^{2x}} dx$$ How do I do it, at first, I thought I treat 2 as $e$ and I will get something like $\dfrac{1}{2} 2^{2x}$, but according to WolframAlpha its supposed to be $\dfrac{4^x}{\lg{4}}$
Jiew Meng
  • 4,593
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How to find $F = \int_a^\infty \frac{e^{-x}}{\sqrt{x-a}}\,dx$

Is there an analytic expression for the following integral? \begin{equation} F = \int_a^\infty \dfrac{e^{-x}}{\sqrt{x-a}}\,dx \end{equation}
Maziar Noei
  • 217
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integration of fractions

i am trying to integrate following equation $$ \int\frac 1{(x^2-1)\cdot (x+2)}\,dx$$ i can represent $(x^2-1)=(x-1)(x+1)$ so,it would be converted in the following form $$\int\frac1{(x^2-1)(x+2)}\,dx=\int \frac1{(x-1)(x+1)(x+2)}\,dx$$ or it is…
dato datuashvili
  • 9,194
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Find the SA of a torus

I have been trying to find the surface area of the torus generated by the rotation of $(x-R)^2 + y^2 = r^2$ about the y axis. I tried to use the equation: $$\int_a^b2\pi y\sqrt{1+\left(\frac {dy}{dx}\right)^2}dx$$ I know that the derivative of the…
TanMath
  • 558
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Help with $\int^{\pi/2}_0 \left( \int_{\pi/2}^y \frac{\sin(x)}{x} dx\right) dy.$

Sub-problem, $\int^{\pi/2}_y \frac{\sin x}{x} dx $, emerging on the page 941 p4b here which asks us to find : $$\int^{\pi/2}_0 \left( \int^{\pi/2}_y \frac{\sin x}{x} dx \right) dy .$$ My instructor once showed me some nice deduction for the thing…
hhh
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