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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What is going on with this integral $\int \frac{dx}{\sqrt{e^{2x} - 9}}$?

A few days ago, I was tasked to solve this integral: $$ \int \frac{dx}{\sqrt{e^{2x} - 9}} $$ The way taught was to recongize the integral as an arcsecant integral. I just can't wrap my head around how it can be arcsecant? The way I did it, which…
VJZ
  • 551
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Issue with integrating double integrals

I am trying to show that $\int_{0
dante
  • 353
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Is there some trick/intuitive/quick way to solve this integral equation?

Suppose that we have two functions $f(x)=f_1(x)+...+f_n(x)$ and $K(x,t)$. If we know that $$F_2(x)+\int_a^b K(x,t)F_1(t)dt=0$$ , where $F_1(x)+F_2(x)=f(x)$. What is the best way to split $f(x)$ into two parts $F_1(x)$ and $F_2(x)$ such that the…
user3741635
  • 406
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Integrate $\int{ \frac{z - Ru}{(R^2 + z^2 - 2Rzu)^{3/2}} du }$

This integral comes from a physics book when calculating a field of an uniformly charged sphere (without Gauss' Law). It says that it can be done by partial fractions, but I cannot imagine how.
pygabriel
  • 147
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The volume of the larger region cut from the solid sphere $x^2 + y^2 +z^2 = 4 $ by the plane z= 1?

Solution using double integrals: I'm trying to calculate this using triple integrals in spherical coordinates. I'm using the method described here Triple Integrals in spherical coordinates . Is this integral and especially the limits correct?…
arvind
  • 178
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Double integrating over quadrilateral region: is it possible to write as a single integral?

Consider the region $R$ which is the region bounded between the four points $$ (0,b) \ , \ \ (a,a+b)\ , \ \ (a,c) \ , \ \ (0,c) $$ where $0
QuantumEyedea
  • 2,805
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Evaluating $\int e^{x/y}dy$.

Find $\int e^{x/y} dy$. Now this is part of an incredibly long exercise but I just got stuck in here, and I feel weird because I don't know how to solve this.
mathobs
  • 49
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How to evaluate the integral $\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx$

Evaluate the intergal: $$\int_{-\pi}^{\pi} \arctan(\pi^x)\,dx.$$ Thank you
alora
  • 31
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How to integrate $\int \frac{dz}{\sqrt{(z^2+1)^3}}$

So I have this integral $$\int \frac{1}{\sqrt{(z^2+1)^3}}dz$$ I tried substituting $z^2+1=t$ but I just get a more complicated integral. WolframAlpha solved the integral really quickly by substituting $z=\tan{u}$, by which the integral transforms…
l0ner9
  • 623
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Find $\int_{0}^{1}\frac{x^2e^{\arctan{x}}}{\sqrt{x^2+1}}\,dx$

$$I=\int_{0}^{1}\frac{x^2e^{\arctan{x}}}{\sqrt{x^2+1}}\,dx $$ First, i noticed that $$ I=\int_{0}^{1} xe^{\arctan{x}}\sin(\arctan{x})\,dx.$$ Using the substitution $\arctan{x}=t$, we get that$$I=\int_{0}^{1}\frac{\sin^2{t}}{\cos^3{t}}e^t\,…
Jack
  • 581
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Prove the Equality of Two Integrals

This is what I've done so far: $V_1 = \pi\int_0^af(x)^2dx = -\pi\int_0^by^2 (1/f'(x))\ dy = -\pi\int_0^by^2 (1/f'(g(y)))\ dy = -\pi\int_0^by^2 g'(y)\ dy$ Integrating by parts: $u=y^2,\ du=2y\ dy, \ v=g(y), \ dv = g'(y)\ dy$ $y^2g(y)|_0^b…
Charles Khunt
  • 233
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Which solution of this integral $\int{\frac{x^2+1}{x^4-x^2+1}}dx$ is correct?

$$\begin{align} \int{\frac{x^2+1}{x^4-x^2+1}}dx&=\int{\frac{1+\frac{1}{x^2}}{x^2-1+\frac{1}{x^2}}}dx\\ &=\int{\frac{1+\frac{1}{x^2}}{(x-\frac{1}{x})^2+1}}dx\\ &=\int{\frac{1}{u^2+1}}du…
Dan Li
  • 93
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Solve $\int\frac{a^2-x^2}{\sqrt{(a^2-x^2)^2-e^2}}dx$

Solve $\int\frac{a^2-x^2}{\sqrt{(a^2-x^2)^2-e^2}}dx$ First I thought of adding and subtracting $e^2$ in the numerator but then realized there was a whole square of $a^2-x^2$ in the denominator, so, dropped this idea. Then I tried substituting…
aarbee
  • 8,246
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Computation of $\int_{0}^{\infty}\frac{x}{\left(1+y^2x^2\right)\left(1+a^2x^2 \right)}dx$

The following integral is simple to compute $$I=\int_{0}^{\infty}\frac{\arctan(x)}{1+x^2}dx \,\,\tag{1}$$ letting $$u=\arctan x \Rightarrow du=\frac{dx}{1+x^2}$$ $$\int\frac{\arctan(x)}{1+x^2}dx=\int u \, du =…
Ricardo770
  • 2,761
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How to solve this integral $\int^{\infty }_{0} {\frac{x \log x}{(1+x^2)^2}} \, dx$?

I will be grateful if you would write me a solution procedure for this integral $$\int^{\infty }_{0} {\frac{x \log x}{(1+x^2)^2}} \, dx. $$ I am sure that an antiderivative is $$\frac{1}{4} \left( \frac{2x^2 \log x}{1+x^2}- \log(1+x^2)…
Anakin
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