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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$\int \frac{dx}{x^3+x^2\sqrt{x^2-1}-x}$

solve: $$\int \frac{\mathrm dx}{x^3+x^2\sqrt{x^2-1}-x}$$ I tried: $$\begin{align}\int \frac{\mathrm dx}{x(x^2-1)+x^2\sqrt{x^2-1}}&=\int \frac{\mathrm dx}{x(\sqrt{x^2-1}\sqrt{x^2-1})+x^2\sqrt{x^2-1}}\\&=\int \frac{\mathrm…
Aligator
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Solve: $\int_{2}^{3}\ \left(\sqrt {2x -\sqrt{5\left(4x-5\right)}}+\sqrt {2x +\sqrt{5\left(4x-5\right)}}\right)dx $

How to solve following integration? $$\int_{2}^{3}\ \left(\sqrt {2x -\sqrt{5\left(4x-5\right)}}+\sqrt {2x +\sqrt{5\left(4x-5\right)}}\right)dx $$
kalpeshmpopat
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Calculate $\int 5^{x+1}e^{2x-1}\,dx$

How to calculate following integration? $$\int 5^{x+1}e^{2x-1}dx$$
kalpeshmpopat
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Integrate $\lvert \int_{a_1} ^{a_n} (x-a_1)^{b_1}(x-a_2)^{b_2}\cdots(x-a_n)^{b_n} dx \rvert $

I know that $${\left\lvert \int_\alpha ^\beta (x-\alpha)^m(x-\beta)^ndx \right\rvert} $$ can be integrated to $${{n!m!(\beta-\alpha)^{m+n+1}}\over {(n+m+1)!} }$$ using recurrence relation. Then, can the generalized form: $$\left\lvert \int_{a_1}…
Verthele
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Concept of $d\left(\frac{1}{T}\right)$

In thermodynamics, I was trying to solve the following integral. $$\int_{T_{1}}^{T_{2}} d \ln K=-\frac{\Delta H}{R} \int_{T_{1}}^{T_{2}} d\left(\frac{1}{T}\right)$$ $$\ln K\left(T_{2}\right)-\ln K\left(T_{1}\right)=-\frac{\Delta…
vik1245
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Evaluate $\frac{1}{2\pi}\int _{-\pi}^\pi \frac{e^{-i\varphi}}{1-k\cos (\varphi)} \, \mathrm{d}\varphi$

I couldn't evaluate this integral. Could you please help me? $$\frac{1}{2\pi}\int _{-\pi}^\pi \frac{e^{-i\varphi}}{1-k\cos (\varphi)} \, \mathrm{d}\varphi,\text{ where $k$ is a constant}$$
mary
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Evaluating $ \int \frac{\sqrt{1+x}}{x} $

I'm trying to evaluate the following integral: $$\int \frac{\sqrt{1+x}}{x}$$ It seems like that I need to use u substitution and partial fraction decomposition? Any tips/advice on how to solve this one? I can't figure it out.
Bob Shannon
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Evaluate the integral $\int_{0}^{\frac{\pi}{3}}\sin(x)\ln(\cos(x))\,dx$

$$\int_0^{\frac{\pi}{3}}\sin(x)\ln(\cos(x))\,dx $$ $$ \begin{align} u &= \ln(\cos(x)) & dv &= \sin(x)\,dx \\ du &= \frac{-\sin(x)}{\cos(x)}\,dx & v &= -\cos(x) \end{align} $$ $$ \begin{align} \int_0^{\frac{\pi}{3}}\sin(x)\ln(\cos(x))\,dx &=…
Evan Kim
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$\lim_{n \to \infty} n \int_0^1 x^np(x) \, dx=$? , where $p(x)$ is a polynomial

I came across the following problem that says: Let $p(x)=a_kx^k+a_{k-1}x^{k-1}+\cdots+a_0$ be a polynomial. Then $\lim_{n \to \infty} n \int_{0}^{1} x^np(x) \, dx$ equals to which of the following? $1.\quad p(1)$ $2.\quad p(0)$ $3.\quad…
learner
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Can we setup the integral as this?

${\dfrac{d}{dx}\Large\int} _{0}^{sinx} x^2\sqrt t\ \ dt$ as ${\dfrac{d}{dx}\ x^2 \Large\int} _{0}^{sinx} \sqrt t\ \ dt$ if so can i do the same thing for this too but I will endup with dt ${\dfrac{d}{dx}\Large\int} _{x}^{\sqrt x} \dfrac{e^x}{x}\…
user10998700
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An integral Lobachevsky calculated incorrectly $\int_{0}^{\infty}\frac{(e^x-e^{-x})x}{e^{2x}+e^{-2x}+2\cos(2a)}dx$

In a recent lecture a professor told a story about the integral below. Lobachevsky calculated this integral at first time incorrectly. Following the publication of the integral, Ostrogradsky sent a letter with correct answer to Lobachevsky. What…
Martin Gales
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Find an implicit solution to $y ^ { \prime } ( x ) = \frac { x ^ { 3 } } { 2 y \sqrt { 1 + y ^ { 2 } } }$

$y ^ { \prime } ( x ) = \frac { x ^ { 3 } } { 2 y \sqrt { 1 + y ^ { 2 } } }$ $\frac { d y } { d x } = \frac { x ^ { 3 } } { 2 y \sqrt { 1 + y ^ { 2 } } }$ $\frac { d x } { d y } = \frac { 2 y \sqrt { 1 + y ^ { 2 } } } { x ^ { 3 } }$ $\int x ^ { 3 }…
Tariro Manyika
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A simple integral with one question

Question is: For $x$ equals $4$ and $9$, why is $t$ not $\pm2$ and $\pm3$ but just $2$ and $3$ ?
Petra
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Why don't terms in the summation expression for an integral (almost) cancel out?

If I think of an integral, $\int_a^b f(x) dx$ as roughly $\sum f(x)\delta x$ where $\delta x$ is very very small, then can't I write this sum as $$ f(x)x - f(x)(x-\delta x) + f(x-\delta x)(x-\delta x) +\dots -f(a)a $$ Then, since $\delta x$ is so…
user106860
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Solve $\int_0^s \left[ 1-\int_{-\infty}^{\infty} \frac{e^{-\frac{y^2}{2}}}{\sqrt{2 \pi}} \tanh\left( x-\sqrt{x}y \right)dy \right] dx$

Let $$ f(x) = 1-\int_{-\infty}^{\infty} \frac{e^{-\frac{y^2}{2}}}{\sqrt{2 \pi}} \tanh\left( x-\sqrt{x}y \right)dy. $$ I would like to understand how to get to the solution of the following integral: $$ g(s)=\frac{1}{2}\int_0^s f(x) dx =…
Enzo
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