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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A quicker method of proving $\int_{0}^{1}{6x(x-1)(x+2)\over (x+1)^3}\ln(x)dx=(\pi-3)(\pi+3)$

$$I=\int_{0}^{1}{6x(x-1)(x+2)\over (x+1)^3}\ln(x)dx=(\pi-3)(\pi+3)\tag1$$ $$I=\int_{0}^{1}\left(6-{12\over 1+x}-{6\over (1+x)^2}+{12\over (1+x)^3}\right)\ln(x)dx\tag2$$ Recall $$\int_{0}^{1}{\ln(x)\over…
gymbvghjkgkjkhgfkl
  • 8,090
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Prove $\int_{0}^{\infty}e^{-nx}\sin^{2k}xdx={(2k)!\over n\prod_{j=1}^{k}(n^2+4j^2)}$

$$I=\int_{0}^{\infty}e^{-nx}\sin^{2k}xdx={(2k)!\over n\prod_{j=1}^{k}(n^2+4j^2)}\tag1$$ Recall $$\sin^{2k}(x)={1\over 2^{2k}}{2k\choose k}+{2\over 2^{2k}}\sum_{j=0}^{k-1}(-1)^{k-j}{2k\choose j}\cos[(2k-2j)x]$$ $$I={1\over 2^{2k}}{2k\choose…
gymbvghjkgkjkhgfkl
  • 8,090
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Prove $12\int_{0}^{\infty}{x\over (e^{x^2}+1)(e^{x^2}+3)}dx=\ln(2)$

$$I=12\int_{0}^{\infty}{x\over (e^{x^2}+1)(e^{x^2}+3)}dx=\ln(2)$$ $u=e^{x^2}\rightarrow du=2xe^{x^2}dx$ $x\rightarrow \infty,\, u=\infty$ $x\rightarrow 0,\, u=1$ $$6\int_{1}^{\infty}{2x\over (u+1)(u+3)}\cdot{du\over…
gymbvghjkgkjkhgfkl
  • 8,090
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Evaluation of $\int_{0}^{2}\frac{\arctan(x)}{x^2+x+2}dx$

Evaluation of $\displaystyle \int_{0}^{2}\frac{\arctan(x)}{x^2+x+2}dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{2}\frac{\arctan(x)}{x^2+x+2}dx =…
juantheron
  • 53,015
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Negative sign that appears in integration

I have to solve the following definite integral $$\int_{0}^{4}r^3 \sqrt{25-r^2}dr=3604/15$$ I have tried a change of variables given by $u=\sqrt{25-r^2}$ where then I find that $dr=-u du/r$ and $r^2=25-u^2$. That change of coordinates then give the…
user288183
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3 answers

Exponential integration

I am working for an investment institution and we need to use upper partial moments. To evaluate them, I need to integrate $$ \int_a^\infty x^2 e^{-ax^2} dx $$ with $a>0$. I found this formula online : $$ \int x^2 e^{-ax^2} dx = \frac{1}{4}…
Daniel
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How do I go about solving this?

I have tried substitution, but it is not working for me. $$ \int_0^\pi \frac{dx}{\sqrt{(n^2+1)}+\sin(x)+n\cos(x)}=\int_0^\pi \frac{n dx}{\sqrt{(n^2+1)}+n\sin(x)+\cos(x)}=2 $$ General form of this integral is $$ \int_0^\pi…
user334593
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The integral $\int\ln(x)\cos(1+(\ln(x))^2)\,dx$

Help with a integral calculus please!? The equation is $$\int\ln(x)\cos(1+(\ln(x))^2)\,dx$$ My teacher told me, i have to use substitution? but i can't still solve it. I've been solving this last week but still i can't get the answer, please help…
Tom Jeric Sison
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Prove that $\frac{\sqrt{3}}{8}<\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sin x}{x}\,dx<\frac{\sqrt{2}}{6}$

Prove that $$\frac{\sqrt{3}}{8}<\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sin x}{x}\,dx<\frac{\sqrt{2}}{6}$$ My try: Using $$\displaystyle \sin x\frac{1-0}{\frac{\pi}{2}-0}=\frac{2}{\pi}$$ So we get…
juantheron
  • 53,015
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4 answers

Dealing with integrals of the form $\int{e^x(f(x)+f'(x))}dx$

Integrals of the form $$\int{e^x(f(x)+f'(x))}dx$$ are very common. And I have seen this form appearing in several exam papers.But the problem I face with this particular type of integral is finding what $f(x)$ could be.Its more of a trial and error…
user220382
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$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$

$$\iiint_V \ x^{2n} + y^{2n} + z^{2n} \,dx\,dy\,dz$$ where V is the unit sphere. No information is given about n but I assume it is an integral. All I could think to do was to convert to spherical co-ordinates and use reduction formulae, but I ended…
M.crolla
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Where did I mistake to integrate $I=\int\sqrt{\frac{\sin(x-\alpha)}{\sin(x+\alpha)}} \; dx\; ?$

It was given to integrate $$I=\int\sqrt{\frac{\sin(x-\alpha)}{\sin(x+\alpha)}} \; dx.$$ Attempt: \begin{align}I&= \int\sqrt{\frac{\sin((x+\alpha)-2\alpha)}{\sin(x+\alpha)}}\; dx\\&= \sqrt{\sin 2\alpha}\int\sqrt{\cot 2\alpha - \cot (x+\alpha)}\;dx\;…
user142971
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1 answer

Integration by parts: How to choose the constant which make calculations easier?

The formula of integration by parts is: $$\int u(x)v(x) dx = u(x)V(x) - \int u'(x)V(x) dx$$ Which can be re-written as: $$\int u(x)v(x) dx = u(x)[V(x)+C] - \int u'(x)[V(x)+C] dx$$ where C is a constant. It makes some integration calculations…
Mohamed Mostafa
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Integral and its limit

Evaluate: $$ \int_{0}^{\pi/2}n \left(1-\sqrt[n]{\cos x}\right) \mathrm{d}x$$ Rewriting this as $$I(n)= \int_{0}^{\pi/2}n \left(1-\sqrt[n]{\cos x}\right) \mathrm{d}x$$ and then Differentiating under the Integral Sign with respect to $n$. I was unable…
User1234
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