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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Integration by parts for a double integral.

I have a complicated double integral that has the following form $$I=\int_y\int_x f(x,y) g(x,y) \, dx dy$$ Suppose I know that this integral $$\int_x g(x,y)\, dx = w(y)$$ beacuse it is easier for me to integrate then the whole product…
Tyrone
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How to integrate the nth root of n

I am working on a problem in which there is an integral containing $x^\frac1x$. I have looked at this question for some help: Behavior of the Nth root of N? I have changed $x^\frac1x$ to $e^{ln(x)/x}$ but I still have not found a way to integrate…
Progo
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A Step in the Proof of Green's Theorem

There is a step in the proof of Green's Theorem where we must combine the line integrals of each curve in the same direction. $$\oint\limits_C P(x,y)\,dx = \int_a^b P(x_1,y_1)\,dx + \int_b^a P(x_2,y_2)\,dx$$ $$\oint\limits_C Q(x,y)\,dy = \int_c^d…
nharren
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Calculation of triple integrals like $ \int_{V'} \frac{ \mathbf{r} - \mathbf{r'}}{\mid \mathbf{r} - \mathbf{r'} \mid ^3} dV' $, on spherical domain

How could one solve integrals in the form: $$ I(\mathbf{r})= \int_{V'} \frac{ \mathbf{r} - \mathbf{r'}}{\mid \mathbf{r} - \mathbf{r'} \mid ^3} dV' $$ where the domain of integration is the sphere: $$ x'^2+y'^2+z'^2 \leq R^2 ~?$$ I would…
NNec
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Riemann-Stieltjes Integration problem

I have two functions $f$ and $g$ and I need to show that $f$ is Riemann-Stieltjes integrable with respect to $g$. I was able to calculate the integral, but I'm not sure how to actually prove why it is Riemann-Stieltjes…
samsonite
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How to integrate $\frac{1}{x\sqrt{1+x^2}}$ using substitution?

How you integrate $$\frac{1}{x\sqrt{1+x^2}}$$ using following substitution? $u=\sqrt{1+x^2} \implies du=\dfrac{x}{\sqrt{1+x^2}}\, dx$ And now I don't know how to proceed using substitution rule.
Banana
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Integration of fraction

Are there any special cases that make the following true $$\int\frac{f(x)}{g(x)} dx = \frac{\int f(x)\ dx}{\int g(x) \ dx}$$ Thanks
Tyrone
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Lebesgue Integration

Can you help me with the following question? Suppose $g \in L^1 ([0,\infty))$ and $\displaystyle\int_0^\infty g(x)dx =1$. Prove that if $f:[0,\infty)\rightarrow \mathbb{R}$ is a continuous function then $n \displaystyle\int_0^1 f(x+t) g(nt) dt…
Berkheimer
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How to integrate $\int \frac{dx}{\sqrt{ax^2-b}}$

So my problem is to integrate $$\int \frac{dx}{\sqrt{ax^2-b}},$$ where $a,b$ are positive constants. What rule should I use here? Should substitution be used or trigonometric integrals? The solution should be:…
jjepsuomi
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Integral $\int\frac{1}{x\sqrt{1-(\ln x)^2}}dx$

It surely should be solved using integration by parts, but I don't know how to proceed: $$ \int\frac{1}{x\sqrt{1-(\ln x)^2}}dx $$ I tried having $du=1/x$ and thus $u=\ln x$, but then I don't know how to deal with the $(\ln x)^2$ in the denominator.…
user159527
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$\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$

I need to evaluate $$\displaystyle\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\left(2x^2+2xy+2y^2\right)} dx\,dy\,$$ I think I'll need $\displaystyle\int^{\infty}_{-\infty} e^{-x^2} dx\,=\sqrt{\pi}$ . But I'm not able to apply it properly.
Mathronaut
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Trigonometric Integration with Negative Exponents

How do you integrate $\csc^4 x/\cot^2 x$? I know that this is the same as $\csc^4 x \cot^{-2} x$ and when you use techniques in trig integrals you end up with $$\int \csc^2 x \csc^2 x \cot^{-2} x \,dx = \int \left(1 + \cot^2 x\right)\left(\cot^{-2}…
sheila
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How do I calculate the area $\int_0^2 \frac{1}{\sqrt{2x-x^2}} dx$?

I have the integral $$\int_0^2 \frac{1}{\sqrt{2x-x^2}}dx$$ I know the answer is $\pi$ but I have problems with limits $0$ and $2$. How can I use $\varepsilon$ in this problem?
USB_MAT
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Integration involving square root function

First , i multiply numerator and denominator by (2-x-x^2)^(1/2), then I split the integral into 2 parts , using trig substitution , part 2 is easy to be integrated .. but when I tried to integrate part 2 , I was completely stuck, I tried trig…
user143997
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How do I integrate: $\int\sqrt{\frac{x-3}{2-x}} dx$?

I need to solve: $$\int\sqrt{\frac{x-3}{2-x}}~{\rm d}x$$ What I did is: Substitute: $x=2\cos^2 \theta + 3\sin^2 \theta$. Now: $$\begin{align} x &= 2 - 2\sin^2 \theta + 3 \sin^2 \theta \\ x &= 2+ \sin^2 \theta \\ \sin \theta &= \sqrt{x-2} \\ \theta…
user63762453
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