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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Calculating $\int_0^\pi \frac{1}{a+b\sin^2(x)} dx $

How do I calculate this integral? $a \gt b$ is given. $$\int_0^\pi \frac{1}{a+b\sin^2(x)} dx $$ I am confused since WolframAlpha says one the one hand, that $F(\pi) = F(0) = 0 $ , but with some random values it isn't 0. What am I missing? Note…
Christian
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What do these symbol mean?

I always see these symbols and others like it when looking at really advanced maths. I have yet to learn anything about it. I was wondering if someone could explain briefly what they are used for. $$\oint \quad\iint \quad \iiint$$
Terry Chao
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Integrate with $-d(x/y)$

Here's an integral which I encountered that uses some unfamiliar notation for me: $$\int-\frac{d(x/y)}{\sqrt{1+(x/y)^2}}$$ What does this mean? I don't have much of an idea. Edit: This problem is from a book on differential equations ($y$ is a…
Tdonut
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Double integral over a parallelogram

I understand the general concept behind double integrals but do not understand how to change the coordinates linearly, and what to do from there. Find $$\int\int_P(x+y)dxdy$$ Where $$P$$is parallelogram with vertices $(0,0), (3,2), (2,4) ,(5,6)$…
fluteman24
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Show $ I = \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = \frac{\pi}{\sqrt 2}$

Show $$ I = \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = \frac{\pi}{\sqrt 2}$$
Kirthi Raman
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show that integral $\int_{0}^{\infty}y\,dx=\frac{\pi^2}{4}$

Given $\sinh{(x)}\sinh{(y)}=1$, I have to find the integral: $$\int_{0}^{+\infty}y\,dx=\dfrac{\pi^2}{4}.$$ I try to use the fact that this $$(e^x-e^{-x})(e^y-e^{-y})=4$$ut i have no idea of how to get $\int ydx=?$
user225250
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Double Integrals Calculus 3

Use a double integral to find the volume of the solid in the first octant bounded by the surfaces: z = xy, z = 0, y = x and x = 1. I did $$\int_0^1 \int_0^x (xy) \ dy\ dx$$ $$= \int_0^1 \frac12xy^2\bigg|_{y=0}^x \,dx = \int_0^1 \frac12x^3 \, dx =…
Solenn
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Is it true that $\int\frac{1}{x}\,dx=\ln x\implies\ln x = \frac{x^0}{0}$?

I have recently learned that $\int\frac{1}{x}\,dx=\ln x$. However, by the power law of integration, the integral of $1/x$ is equal to $x^0 / 0$ which is undefined. Therefore, is $x^0 / 0 = \ln x$? Am I correct in this assumption?
Gigabit
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Evaluation of Painful Integral:

I'm trying to find a way to unit test a numerical evaluation of: \begin{align} I(y) := \int_{0}^{y} \frac{\sin(x)}{x} \frac{1}{\sqrt{1+ax}} \, \mathrm{d}x, \qquad a \in \mathbb{R} \end{align} Is there a nice representation of this function in terms…
user14717
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How to integrate $\int \frac{1}{g-kv^2}~dv$

How to integrate $$\int \frac{1}{g-kv^2}~dv$$ where $v$ is velocity and $g,k$ are known constants. I have been stuck for a while on this and have now turned to SE for help any suggestions?
Mechin
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Evaluating $\int\frac{1}{5\cos x+\sin x+7}~dx$

Evaluating $$\int\frac{1}{5\cos x+\sin x+7}~dx.$$ This can be done by substituting $$\sin x = \frac{2t}{1 + t^2}$$ and $$\cos x = \frac{1 - t^2}{1 + t^2}.$$ However after I substitute it I cannot simplify it to get anything easier to…
user164612
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Having trouble solving $\int\frac{5x^2+3x+2}{x(x+1)^2}$

I've first transformed the integral to $$\int\frac{5x^2+3x+2}{x(x^2+2x+1)}dx$$ Which gave…
student
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why does integration by parts give a different answer

so i know the answer for $\int{\frac{x \, dx}{(x+b)^{2}}} \quad \textrm{is} \quad \frac{b}{x+b} + ln|x+b| $ But i tried integration by parts and obtained the following, Setting $ u= x, \, du=dx, \, dv = \frac{dx}{(x+b)^{2}}, \, v=\frac{-1}{x+b} $…
user2804865
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A troublesome integral $\frac{1}{a(2\pi)^2}\int^{\infty}_{0} \frac{r(b-cr)}{\sqrt{r^2+m^2}}\sin{(ar)}e^{-i\sqrt{r^2+m^2}t-nr} dr$

I've been trying to solve this integral without much luck. $$\frac{1}{a(2\pi)^2}\int^{\infty}_{0} \frac{r(b-cr)}{\sqrt{r^2+m^2}}\sin{(ar)}e^{-i\sqrt{r^2+m^2}t-nr} dr,$$ or alternatively $$\frac{1}{a(2\pi)^2}\int^{\infty}_{0}…
Rammus
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Example of Why Integration Order Is Not Always Interchangable

I am being asked to find $\int_1^\infty\int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2}\,dx\,dy$ and $\int_1^\infty\int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2}\,dy\,dx$. I am even told to "notice that" $\int_1^\infty…
fullyhip
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