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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What is $\int \frac{x}{x^5-7} dx$?

I have tried out many trigonometric substitution like $x=\sin^{\frac{2}{5}}z$. But it did not work.
Aman
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GRE Integration Problems $ \int_{-\pi/4}^{\pi/4} (\cos(t) + \sqrt{1+t^2} \sin(t)^3\cos(t)^3) dt$

I recently took a practice GRE and I noticed that the integrals on the test were not too advanced or laborious, so long as one noticed the "trick." For instance, the following integral: $$ \int_{-\pi/4}^{\pi/4} (\cos(t) + \sqrt{1+t^2}…
msm
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Evaluating the integral $\int_{0}^{1}\frac{x^n}{1 + x^n}\,\mathrm dx$.

For each positive integer $n$, let $f_{n}$ be the function defined on the interval [0,1] by $$f_{n}(x) = \frac{x^n}{1 + x^n},$$ I want to find $$\int_{0}^{1}f_{n}(x)\,\mathrm dx,$$ I thought that I could add and subtract $1$ to the numerator, but…
Emptymind
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How to integrate $\int x^2 \sin^2(x)dx$

I don't know how to integrate $\int x^2\sin^2(x)\,\mathrm dx$. I know that I should do it by parts, where I have $$ u=x^2\quad v'=\sin^2x \\ u'=2x \quad v={-\sin x\cos x+x\over 2}$$ and now I have $$ \int x^2\sin^2(x)\,\mathrm dx = {-\sin x\cos…
Emilia
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How to evaluate the integral:$l(y)=\int\limits_\beta^\infty \theta\exp(-y\theta)\alpha\exp(-\alpha\theta)\, d\theta$ where $\alpha,\beta,\theta,y>0$

how to evaluate this integral: $$l(y)=\int\limits_\beta^\infty \theta\exp(-y\theta)\alpha\exp(-\alpha\theta) \, d\theta$$ where $\alpha,\beta,\theta,y>0.$ Because I find it infinity! Can anyone help me to evaluate this integral? Thank you.$$$$ I…
gips
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Integrate over a discontinuity

Evaluate $$\int_{0}^{1} \dfrac{1}{1 + \left(1 - \dfrac{1}{x}\right)^{2015}} dx$$ Wolfram says it's 1/2 but I thought that you can't integrate this because there is a discontinuity at x = 0. Progress: The inverse substitution $x$ = $\dfrac{1}{u+1}$…
user245640
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Finding Trigonometric Integration $\int\frac{\sin^8 x+\cos^9 x}{1+\tan^9 x}dx$

Finding $\displaystyle \int\frac{\sin^8 x+\cos^9 x}{1+\tan^9 x}dx$ $\bf{Attempt:}$ Integral $\displaystyle I = \int\frac{(\sin^8 x+\cos^9 x)}{\sin^9 x+\cos^9 x}\cdot \cos^9 xdx $ Put $(\sin^9 x+\cos^9 x) = t,$ then $9\sin x\cos x(\sin^7 x+\cos^7…
DXT
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Show that $\int _0^1\frac{\ln \left(1+\left(\frac{1-t}{1+t}\right)^2\right)}{t\left(\ln t\right)^2}dt=\ln 2$

According to WolframAlpha the integral $$\int _0^1\frac{\ln \left(1+\left(\frac{1-t}{1+t}\right)^2\right)}{t\left(\ln t\right)^2}dt$$ is shown to have a decimal expansion exactly identical to $\ln 2$. How can we prove they are equal? This integral…
tyobrien
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Determining bounds of integration given a set and a set function?

Given the set $$C=\{(x,y,z,) : x^2+y^2+z^2 \leq 1 \}$$ Using spherical coordinates, evaluate the function $$Q(C)=\iiint_C \sqrt{x^2+y^2+z^2} \, dx \, dy \, dz$$ So... I can see that the function easily converts to the triple integral of $\rho$. My…
rocksNwaves
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Evaluate an indefinite integral

Find the value of $$\int{\frac{x^2e^x}{(x+2)^2}} dx$$ My Attempt: I tried to arrange the numerator as follows: $$ e^xx^2 = e^x(x+2-2)^2 $$ but that didn't help. Any guidance on this problem will be very helpful.
MathsLearner
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Evaluating $I=\int_{-\infty}^\infty \frac{u^2}{5u^2\left(u^2+1\right)+2}\,\mathrm{d}u$?

This integral popped up when I was trying to solve this integral : $$\int_{-\pi/2}^{\pi/2}\frac{\sin^2x}{4(\cos^4x+2\sin^4x)+2\sin^2 2x}\,dx $$ I simplified it a little bit, substituted $\tan(x)=u$ and came up with, $$I= \int_{-\infty}^\infty…
Mathoholic99
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Question regarding integration by substitution.

The theorem on integration by substitution says that $$\int_{\phi(a)}^{\phi(b)}f(x)dx=\int_{a}^{b}f(\phi(t))\phi'(t)dt$$ provided that $\phi$ has an integrable derivative. My question is, shouldn't $\phi$ be monotonic on $[a,b]$? I have this doubt…
john
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finding value of g(2) using integration

Let g:$[0,\infty) \to\mathbb{R}$ be continuous function satisfying $\int_{0}^{x^2(1+x)} g(t)dt=x , for all x\in[0,\infty)$. Then g(2) is equal to ? given : $\int_{0}^{x^2(1+x)} g(t)dt=x $ differentiating both sides with respect to x…
dipali mali
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Evaluate $\int_1^{\infty} {\frac{\ln{x}}{(x-1)(2x-1)}\,dx}$

Evaluate: $$\int_1^{\infty} {\left(\frac{\ln{x}}{\left(x-1\right)\left(2x-1\right)}\,dx\right)}$$ It turns out that the value of the integral is exactly: $$\frac{1}{12}\left(\pi^2+6\ln^2{2}\right)$$ as found by WolframAlpha, but Wolfram gives no…
Ant
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How to solve this absolute value equation

I have the following integral $$\int_0^{\pi/2} |\sin x-\cos x|dx. $$It's a simple integral but when I try to solve the module I get stuck. I took $\sin x-\cos x>0$ and squaring this I found that $\sin 2x<1$. When I apply $\arcsin$ it would mean…
Lola
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