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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Evaluate $\int \sqrt{x^2 + a^2}\mathrm{d}x$

I tried using integration by parts and could get to this point: \begin{align} \int \sqrt{x^2 + a^2}\mathrm{d}x &= x\sqrt{x^2+a^2} - \int x\frac{2x}{\sqrt{x^2 + a^2}}\mathrm{d}x \\\\ &= x\sqrt{x^2+a^2} -2\int \sqrt{x^2 + a^2}\mathrm{d}x + \int…
techie11
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Solving an integral when using a dummy variable

I'm trying to solve an integral and I don't remember how to when a variable is in the integral symbol, if that makes sense. I'm not sure what the correct terminology is. So say I have an integral like: $$f(x) = \int_{-3}^{x^{3}} t^2 dt$$ How do I…
InBetween
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A question from Integration Bee 2022: Evaluate $\int^2_{1/2}\ln(\frac{\ln(x+\frac{1}{x})}{\ln(x^2-x+\frac{17}{4})})dx$

I am trying to evaluate the integral $$\int^2_{\frac{1}2}\ln\left(\frac{\ln(x+\frac{1}{x})}{\ln(x^2-x+\frac{17}{4})}\right)dx$$ by separating it as…
HeyFan
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Calculate double integral $\int_Se^{\frac{x}{y}}dxdy$

Calculate double integral $\int_Se^{\frac{x}{y}}dxdy$ for the region $1 \le y \le2$ and $y \le x \le y^3$ What I have tried: $y = 1, y=2 \\ x = y, x=y^3$ $1 \le x \le 2, \text{when }x=y \\ 1 \le x \le 2^{\frac{1}{3}}, \text{when }x=y^3 \\ 2 \le x…
Working dollar
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Evaluating $\iint_D\frac{\sin y}{y}dxdy$, where $D$ is the region enclosed by $y=2$, $y=1$, $y=x$, $y=2x$

I have been given this exercise: Calculate the double integral: $$\iint_D\frac{\sin(y)}{y}dxdy$$ Where $D$ is the area enclosed by the lines: $y=2$, $y=1$, $y=x$, $2y=x$ (not $y = 2x$). Visualising $D$ is easy. You can split D in two sub areas and…
Giannis Koulouras
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Evaluate $\int_2^\infty{\frac{3x-2}{x^2(x-1)}}$

To be shown that $\int_2^\infty{\dfrac{3x-2}{x^2(x-1)}}=1-\ln2$ My thought: $\dfrac{3x-2}{x^2(x-1)}=\dfrac{3x}{x^2(x-1)}-\dfrac{2}{x^2(x-1)}$ • $\dfrac{3x}{x^2(x-1)}=\dfrac{3}{x(x-1)}=\ldots=-\dfrac{3}{x}+\dfrac{3}{x-1}$ •…
darkchampionz
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Is it possible to eliminate $g(x)$ in the formula $\dfrac{\int g(x)a(x)}{\int g(x)b(x)}$

I am hoping someone can show me how to do the following: I have two integrals containing a function $g(x)$ that I would like to eliminate. The formula looks like this. $\frac{\int g(x)a(x)}{\int g(x)b(x)}$ Is there any way to eliminate the $g(x)$…
Paul
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Evaluate $\int_{0}^{\pi/2} \frac{y \sqrt{\cos(y)}}{\sqrt{\cos(y)} + \sqrt{\sin(y)}} dy$

This integral: $$\int_{0}^{\pi/2} \frac{y \sqrt{\cos(y)}}{\sqrt{\cos(y)} + \sqrt{\sin(y)}} dy$$ came up when I was playing around with trying to solve this similar problem: Is there a closed form for $\int_0^{\pi/2} \dfrac{e^{-x}\sqrt{\cos…
The_Sympathizer
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How prove this integral $\int_{0}^{1}\frac{x^m\ln{x}}{x-1}dx=\sum_{r=m+1}^{\infty}\frac{1}{r^2}$

prove that $$\int_{0}^{1}\dfrac{x^m\ln{x}}{x-1}dx=\sum_{r=m+1}^{\infty}\dfrac{1}{r^2}$$ This true? my idea let $$I(m)=\int_{0}^{1}\dfrac{x^m}{x-1}dx\Longrightarrow I'(m)=\int_{0}^{1}\dfrac{x^m\ln{x}}{x-1}dx$$ and we must…
math110
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Integration and derivative

I want to compute the derivative $$\frac{\mathrm{d}}{\mathrm{d}x}\left(\int_{x^3}^{e^{2x}}\frac{\sin(3y)}{\sqrt{y^2+\cos(3y)+1}}\,\mathrm{d}y\right). $$ Can anyone give me some hints for this question? It is too complex and I don't know what is the…
Alex
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Calculating the Area when inverse function is given

For all the real number, $f\left( x \right)$ is an increasing function that is differentiable while satisfying the following conditions? (A) $f\left( 1 \right) = 1,\int\limits_1^2 {f\left( x \right)dx} = \frac{5}{4}$ (B) Say g(x) is an inverse…
Samar Imam Zaidi
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An unusal integral question

I'm stuck on this question of evaluating an integral of the type: $$\int \dfrac{x^{2n+1}}{x^{2k+1}+1} \,dx$$ with $k > n$. More concretely I try to compute: $$\int_1^2 \dfrac{x^{3}}{x^{5}+1} \,dx.$$ Firstly, I split $x^5 + 1$ into parts and tried…
ANN_Techy
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Find the integral $\int_{0}^{\infty}\frac{ e^{-ax^{2}}-e^{-bx^{2}} }{x}dx,\ a>0,\ b>0$

Find the integral $$\int_{0}^{\infty}\frac{ e^{-ax^{2}}-e^{-bx^{2}} }{x}dx,$$ $a>0,\ b>0$. I tried to split the integral and use Feynman substitution, but I got an answer that doesn't look like the right one. I considered this integral…
Maksim S
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Analytic solution of $ \int_t^{t+r} \frac{x^2}{(t+r-\sqrt{2rx-x^2})^4}dx $

Could anyone help me to solve this defined integral analytically? $$ \int_t^{t+r} \dfrac{x^2}{(t+r-\sqrt{2rx-x^2})^4}dx $$
Giorgio Pia
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Strange integral behaviour

If I take the following integral: $$\int_0^l \sin\left(\frac{n\pi x}{l}\right)\sin\left(\frac{m\pi x}{l}\right)\mathrm dx, \tag{1}$$ and I put it into an integral calculator, it produces the following formula: $$y=-\frac{l((n-m)\sin(\pi n+\pi…
Charlie
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