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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Measure zero implies volume zero when volume is defined

I need to prove that if a set has measure zero and it has well-defined volume, then its volume is zero. I have tried to bound the lower sums of the indicator function, but I'm stuck. Given a zero-measure set $A$ which volume is well-defined, if $R$…
Seven
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how to evaluate these integrals?

1) $\displaystyle \int_{-\pi}^\pi \frac{\sin^3(x)dx}{x^2}$ 2) $\displaystyle \int_{0}^{\pi/2} \frac{\cos^3(x)\sin(x)dx}{1 + \cos^2x}$ I have no ideas how to deal with it. I tried to evaluate them by parts and use change of variables, but nothing…
Michael
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interesting integral involving powers of tan in the denominator

I ran across this integration problem that has an interesting pattern. $$\int_{0}^{(n-1)\pi}\frac{1}{\tan^{n}(x)+1}dx=\frac{(n-1)\pi}{2}$$ I evaluated increasing values of n up to n=10, and the result is always one half the upper limit of…
Cody
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Find integration with limit from 1 to 2

Find the integration my try : Unable to solve further
cattt
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Is $\int\limits_{-a}^{0} f(x)\,\mathrm{d}x = \int\limits_0^a f(-x)\,\mathrm{d}x$ true?

Is the following theorem true? If yes, how to prove it? $$\int\limits_{-a}^{0} f(x)\,\mathrm{d}x = \int\limits_0^a f(-x)\,\mathrm{d}x$$ Update: $$ \int\limits_{\color{red}-a}^{0} f(x) \,\mathrm{d}x = \color{red}-\int\limits_{\color{red}+a}^{0}…
sergej
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what is $\int_0^1 \sqrt{1+2\sin^2(t)+\cos^2(t)}dt$?

The question is : Is $\sqrt 3$ the length of $$\Gamma =\{\gamma (t)=(t,\sin(t),\sqrt 2\cos(t))\mid t\in [0,1]\} \ \ ?$$ So it is $$\int_0^1\|\gamma '(t)\|dt=\int_0^1 \sqrt{1+2\sin^2(t)+\cos^2(t)}dt$$ I tries to do many substitution as…
user621345
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If $a$ and $b$ are constants, calculate the definite integral

$$\int_{-\infty}^{+\infty} F(x-b) f(x-a) dx$$ $$f(x) = \exp(-x-e^{-x}), \qquad x \in (-\infty, +\infty)$$ $$F(x) = \int_{-\infty}^x f(t) dt$$ I calculated already the integral of $F(x)$, which is $\exp(-e^{-x})$, but I am stuck on the other one, I…
user10471408
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Why is $\int d^3r_1 \, d^3 r_2 \, \frac{F(\vec{r}_1) F^*(\vec{r}_2)}{|\vec{r}_1 - \vec{r}_2|}$ positive?

I have the following quantity: $$ I = \int_{\mathbb{R}^3 \times \mathbb{R}^3} d^3r_1 \, d^3 r_2 \, \frac{F(\vec{r}_1) F^*(\vec{r}_2)}{|\vec{r}_1 - \vec{r}_2|} $$ $I$ is obviously real by symmetry. How can I show that it is also positive…
Jolyon
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Need help in solving integral

$$\int\frac{dx}{(z^{2}+x^{2})^{3/2}}$$ I arrived at this after a substitution $t=x/z$: $$\frac{1}{z^{2}}\int\frac{dt}{(1+t^{2})^{3/2}}$$ but now stuck with that 3/2 in the exponent.
Mykolas
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How to solve $\int \frac{a^x + b^x}{c^x + d^x}\:dx$

As an extension of this question how would we address integrals of the form: $$\int \frac{a^x + b^x}{c^x + d^x}\:dx$$ Where $a,b,c,d \in \mathbb{R}^{+}$ are the values are distinct. Does anyone have any starting points?
user150203
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multiplying $\frac{dy}{dx} + \frac{2x+ x^2 + y^2}{2y}=0$ by integrating factor $2ye^x$

I've been introduced to a new concept called the integrating factor and I unsure how it effects the separation of variables for this equation. Can anyone help... By multiplying the following differential equation by $2ye^x$ and carefully…
Ricky Rozay
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Prove the reduction formula $I_n = \frac{1}{n} \cos^{n-1}(x) \sin(x) + \frac{n-1}{n} I_{n-2} \ for \ n \geq 2$

Let $ $$ I_n = \int\cos^n{x} \ dx, \;\text{ for } n=0,1,2,3, \ldots$ Prove the reduction formula $$I_n = \frac{1}{n} \cos^{n-1}(x) \sin(x) + \frac{n-1}{n} I_{n-2}, \; n \geq 2.$$ How do I approach this question? Is there anything I should look out…
Steve
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How can I find $\int(\sin ^4 x ) dx$?

Possible Duplicate: Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$ Hi, My question is: How can I solve the following integral question? $$\int(\sin ^4 x ) dx$$ Thanks in advance.
MAxcoder
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What methods can be used to solve $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $

I'm seeking methods to solve the following definite integral: $$ I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $$
user150203
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Find a function that that makes the value of this improper integral equal to 1.

I have the following integral: $$I(t) = \int_{0}^{t} \sqrt{1- \frac{a(x)^2}{c^2}}dx$$ where $a(x)$ is some continuous function of $x$, and $c$ is a constant. Also $a(x) < c$ for all $x >0$. It can be seen that the solution to the integral is thus…
Kenshin
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