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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How to solve integral $\int \log (x+\frac{1}{2}-\frac{\arctan(\tan(\pi(x+\frac{1}{2})))}{\pi})dx$?

I have been trying to work out how WolframAlpha derived this answer, $$\int \log\left(x+\frac{1}{2}-\frac{\arctan\left(\tan\left(\pi\left(x+\frac{1}{2}\right)\right)\right)}{\pi}\right) {\rm d}x=x \log \left(x+\frac{\tan ^{-1}(\cot (\pi x))}{\pi…
onepound
  • 1,365
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How do I solve this integral via integration by parts when it shows a recursion-like behavior?

I've come across an integral while solving a question for ODE's. The integral is as follows: $$\int e^{-2x}x\,dx$$ From here I set: $$ \begin{align} f(x) = e^{-2x}\quad & f'(x) = -2e^{-2x} \\ g(x) = \frac{1}{2}x^2\quad & g'(x) = x \end{align} $$ and…
Sean
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Integrable or antiderivative

Is there any difference between a function being integrable (since there are many ways to define whether a function is integrable, let's stick to Riemann integration.) and it's antiderivative existing? The antiderivative need not be any…
user59756
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How to maximize the following integral?

Suppose $f : [1,3]\to \mathbb R$ such that $-1\leq f(x)\leq 1, x\in [1,3]$, and $$\int _1^3 f(x)\mathrm{d}x = 0. $$ Determine the largest possible value of $$\int _1^3 \frac{f(x)}{x}\mathrm{d}x. $$ I have found that largest value is $\log3$. But…
vqw7Ad
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Question on integration of $x^{x}$

I want to evaluate $\displaystyle\int x^{x} \, dx$. For this I am taking $$\begin{align} x^{x}&=t\\ \log {x^{x}}&=\log {t}\\ x\log x&=\log t\\ e^{x\log x}&=t\\ \end{align}$$ Then I am stuck! Please make suggestions on how to do this example.
Kns
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Limits of integration for parametric equation

For the picture attached I am wondering why I cannot take the limits from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. If I take those limits, Sine vanishes (in the second last step) and the answer varies significantly. Where am I going wrong?
shubham jain
  • 41
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Integration by parts of delta function

I am having the worst time trying to solve this integral, $$ \int g(t)\frac{d}{df}\delta[f(t)]dt, $$ $$ = \int g(t)\frac{dt}{df}\frac{d}{df}\delta[f(t)]dt. $$ This should yield, $$ -\bigg[ \frac{dt}{df}\frac{d}{dt} \bigg( \frac{g(t)}{\frac{df}{dt}}…
Shinobii
  • 175
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Integrating $\int^2_0 xe^{x^2}dx$

Well what I was thinking was to integrate the indefinite integral first. $u=x^2$, $x=\sqrt u$ $du=2xdx = 2\sqrt {u} dx$ $dx= \frac{1}{2\sqrt{u}}du$ $\int xe^{x^2} dx = \int \sqrt{u}\frac{1}{2\sqrt{u}} du =\frac{1}{2}\int e^u du = \frac{1}{2}e^u…
user477465
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$\int_0^{100}\frac{e^{-x}}{x+100}dx>0.005$?

$\int_0^{100}\frac{e^{-x}}{x+100}dx>0.005$? My attempt: $$\int_0^{100}\frac{e^{-x}}{x+100}dx>\int_0^{100}\frac{e^{-x}}{200}dx=\frac{1-e^{-100}}{200}$$ A little bit error. How to amend it?
xldd
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Indefinite integral of $\int x\sqrt{x-1} \, \mathrm dx$

How can I evaluate the indefinite integral $$\int x\sqrt{x-1} \, \mathrm dx?$$ I tried to calculate it using integration by parts, I get $$\int x\sqrt{x-1} \, \mathrm dx = \frac{2}{3}x(x-1)^{3/2} - \frac{2}{3}\cdot\frac{2}{5}\cdot(x-1)^{5/2}$$ But…
TTomi
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When does a function NOT have an antiderivative?

I know this question may sound naïve but why can't we write $\int e^{x^2} dx$ as $\int e^{2x} dx$? The former does not have an antiderivative, while the latter has. In light of this question, what are sufficient conditions for a function NOT to have…
Rob
  • 622
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Computing $\int_0^{\infty} e^{iax}/(x^2+1)dx$

I am having trouble solving this integral with complex analysis $$\int\limits^{\infty }_{0}\frac{e^{iax}}{x^{2} +1} dx$$ I have tried two different contours; those being contour 2 contour 1 with both contours, I got the answer $\int\limits ^{\infty…
Highvoltagemath
  • 509
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Theorem related with mean value Integral for integrals

Let $f,g: [a,b] \longrightarrow \mathbb{R},$ $f$ is monotone with $f(a)=0$, and $g$ is continuous. Then there is $\theta \in [a,b] $ such that: $$\displaystyle \int_a^b f(x)g(x)dx = f(b)\int_{\theta}^bg(x)dx.$$ My attempt: If I consider that…
Joãonani
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Finding an integral using a table?

Am I correct for pattern matching this integral? I have $$\int \frac{\sqrt{9x^2+4}}{x^2}dx$$ Does this pattern match with: $$\int \frac{\sqrt{a^2 + x^2}}{x^2}dx = -\frac{a^2 + x^2}{x} + \ln(x + \sqrt{a^2 + x^2}) + c$$ If I factor out the 9, I…
Jwan622
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Find the integral $\int \frac{(\ln(x))^2}{x^3} \, dx$

$$\int \frac{(\ln(x))^2}{x^3} \, dx $$ Starting off with Integration by Parts $$ \begin{align} u = \ln(x)^2 &~~~ dv = x^{-3} \\\\ du = 2\ln(x)dx &~~~ v = \frac{x^{-2}}{-2} \end{align} $$ $$ \begin{align} \int \frac{(\ln(x))^2}{x^3} &= (\ln(x))^2…
Evan Kim
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