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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Impossible Integration Problem?

Let $f(x, y) = \sqrt{x + y\sqrt{x + y\sqrt{...}}}$ Evaluate $\int_0^2 \int_0^2 f(x, y) \,dy \,dx$ I gave this problem to a few of my friends and my Calculus teacher and no one could solve it, nor could any online math tool/solver, although the…
Ryan Rudes
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Integrating $ \frac{1}{\mathrm{e}^{2\, t}\, \left(\mathrm{e}^{t} + 1\right)} $

I would like to integrate: $$ \int{ \frac{dt}{\mathrm{e}^{2\, t}\, \left(\mathrm{e}^{t} + 1\right)} } $$ But I don't know where to start. Ideas? The answer according to mupad is $$ t + \frac{1}{\mathrm{e}^{t}} - \frac{1}{2\, \mathrm{e}^{2\, t}}…
bobobobo
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Visibly Unequal Answers in Using Different Methods of Integration

Question: Evaluate the Integral:$\displaystyle\int\sec(x)\,\mathrm dx $ Approach 1 : Write $\sec(x)$ as $\ \frac{1}{\cos(x)}$, $$\ \int \frac{1}{\cos(x)}\,\mathrm dx $$ Substitute $\sin^2(x) + \cos^2(x) $ for $1$, $$\int \frac{\sin^2(x) +…
Prajval K
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Find the integral $\int ^{2\pi}_{0} \sin (20x) \sin x \cos x\sqrt{1+400 \sin^2x}\; dx$

Find the integral $$\int ^{2\pi}_{0} \sin (20x) \sin x \cos x\sqrt{1+400 \sin^2x}\; dx$$ my attempt: Let $1+400 \sin^2x=t $ then $ 800 \sin x\cos x dx=dt$ $\displaystyle\int ^{2\pi}_{0} \sin (20x) \sin x \cos x\sqrt{1+400 \sin^2x}=\int ^{2 \pi}_{0}…
Inverse Problem
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One Step in Proving the Gamma of $1 \over 2$

Good Afternoon All. There is one step in the proof that I never quite understood. Let $I = \int_{0}^{\infty} e^{{-u}^2} du$. Then $$I^2 = \int_{0}^{\infty} e^{{-u}^2} du \int_{0}^{\infty} e^{{-v}^2} dv$$ Now, if I say that since…
Andy Tam
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Problem with integration by substitution.

Using the substitution $u=1-x$, compute the integral of $\int{x(1-x)^2}dx.$ My Work: Let $u=1-x.$ Then $\mathrm dx=-\mathrm du$ and $x=1-u,$ so $$\int{x(1-x)^2}\,\mathrm dx\\ =-\int{(1-u)u^2}\,\mathrm…
Joe Watkins
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Simple generalized integral

The integral to compute is $\displaystyle\int_0^\infty \frac{1}{3+x^2} \ \mathrm dx$. I know how to compute the indefinite integral of this function - I obtained: $$\frac{\sqrt{3}}{3} \arctan\left(\frac{x}{\sqrt{3}}\right).$$ But when I compute the…
The Answer
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Calculate the definite integral

Calculate the following integral: $$ \int^3_{-3}\left(\frac{\arctan(\sqrt{|x|})}{1 + (1+x^2)^x} \right)dx $$ I know that each function can be represented as a sum of even and odd function such that: $$ f(x)=\frac{f(x) + f(-x)}{2} +\frac{f(x) -…
Paul
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How to find the following indefinite integral?

$$ \int {dx \over {\sin^3 x+\cos^3 x}}$$ Can this integral be found by substitution or any other method such as complex number?
DebojyotiMukh
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Solving Friedmann equation for a(t) in quintessence universe

I'm having a tough time with an integral, and I'm not sure if I'm just not thinking about it right. To get full context, here is the first part of the question (and my solution to that first part, as it seems like it will be used in the part I'm…
ObiDonKenobi
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Help required with limit

In finding the asymptotic value of a certain quantity I ended up with the following: $$f(n)=(n-1)B_{1/2}(n,n+2)$$ $$g(n)=(n+1)B_{1/2}(n+2,n)$$ $$ h(n)=4^n \left( f(n)-g(n)\right)$$ Numerical simulations lead me to believe that h(n) approaches 1 as…
AgnostMystic
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The integral of $\frac{1}{1+x^n}$

Motivated by this question: Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$ I got curious about finding a general expression for the integral $\int \frac{1}{1+x^n},\,n \geq 1$. By factoring $1+x^n$, we can get an answer for any given $n$ (in…
Lord Soth
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Gauss law from physics

I would like to apologise for the question that I am going to pose; and for my curiosity. Q: How to evaluate the integral below of the function $E(x, y, z) $over $\Omega$ which is the surface of a ball or sphere. Suppose that the radius of sphere is…
sergei ivanov
  • 194
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Compute difficult integral $\int \frac{dx}{2 + x + \sqrt{1 - x^2}}$

To solve the integral $$I = \int \frac{dx}{2 + x + \sqrt{1 - x^2}}$$ I have tried several things, such as $t = \arcsin x$, because $\cos(\arcsin x) = \sqrt{1 - x^2}$. If I am not wrong, we can conclude with this variable change $$ I = \int…
joseabp91
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Integral of $\int^1_0\frac{dx}{\sqrt{x+3}-1}$

I want to solve this integral and need some directions. $$\int^1_0\frac{dx}{\sqrt{x+3}-1}$$ I decided to call $x+3 = t^2 \rightarrow 2tdt = dx$ then : $$\int^1_0 \frac {2tdt}{t^2-1}$$ Now what should I do? call $t^2-1 = u$ ? and do the same thing…
Ofir Attia
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