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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Calculate $I=\int_0^1\frac{\ln(1+x)}{1+x^2}\,dx$

How can I calculate the integral $I=\int_0^1\frac{\ln(1+x)}{1+x^2}\,dx$ by substituting $t=\frac{1-x}{1+x}$.
pourjour
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Calculating $ \int_{0}^{1}\frac{\sqrt{x}}{\sqrt{1-x^{6}}}\mathrm{d}x $

I can't calculate the Integral: $$ \int_{0}^{1}\frac{\sqrt{x}}{\sqrt{1-x^{6}}}dx $$ any help would be great! p.s I know it converges, I want to calculate it.
Tomer
  • 31
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Is it possible to do this integral?

Is it possible to do this integral? $$\int_{-\infty}^{\infty} \operatorname{sech}^2(Bx)\operatorname{sech}^2(ax) dx,$$ where $B$ and $a$ are constants. I tried using Mathematica, but no dice...
Bob Riley
  • 51
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Integral of an exponent [updated - new way of solving the problem]

I have been struggling with this integral for a whole week now. The integral form is quite simple but extremely difficult for solving it. It related to the heat conduction problem, but I have simplified it so that we could see it a bit more…
Nguyễn Quang Dũng
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Integral question - $\int\frac{(x+6)\,dx}{4x-x^2}$

Integral question - $\int\frac{(x+6)\,dx}{4x-x^2}$ What I did is $$\int\frac{(x+6)\,dx}{x(4-x)}$$ then $$\int\frac{(x+6)\,dx}{4x-x^2}= \int\left(\frac{A \,}{x}+\frac{B}{4-x}\right) dx$$ this is the right way? Thanks! Addition:…
Ofir Attia
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Horizontal Centre of mass of semi circle coming as non-zero for some reason

Centre of mass is defined as, $$ \overline{x} = \int x \rho dA$$ for a semi circle, above the x axis, $$ \overline{x} = \rho \int_{0}^{R} \int_{-\sqrt{1-y^2} }^{\sqrt{1-y^2} } x dx dy$$ This becomes (my origin is at center of semi circle) $$…
tryst with freedom
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Solving two specific integrals with so-called Feynman parameters

I am trying to solve some integrals which appear in the context of renormalization in quantum field theory and integrals with so-called Feynman parameters, but I am unable to reproduce what is (according to the lecture notes) the correct answer. In…
psiarchon
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By methods of differentiation under the integral sign

Could this result be proven by methods of differentiation under the integral sign? I only take interest in differentiation way. $$\int_0^\infty \frac{\ln(\tan^2 (ax))}{1+x^2}dx = \pi\ln(\tanh(a))$$
Kilgore Trout
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If $f(x)=\int_{0}^{x}\cos\frac{1}{t}\ {dt}$, then caculate $f'(0)$

If $f(x)=\int_{0}^{x}\cos\frac{1}{t}\ {dt}$, then calculate $f'(0) $. I found the answer there and understood its solution. But when I solve it myself, I first tried to solve it directly: $$f'(0)=\frac{\int_{0}^{x}\cos\frac{1}{x}dx}{x},$$ And use…
fractal
  • 551
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How to simplify double integral with the upper bound of the inner integral as the integration variable of the outer integral

$$ \begin{split} B(0,T) & = \color{fuchsia}{\text{EV[non-default scenario]}} + \color{blue}{\text{EV[default scenario]}}\\ & = E\left[\color{fuchsia}{\exp\left(-\int_0^T\!\!\! r_t dt\right)·\mathbf{1}_{\{T<\tau\}}} + \color{blue}{\int_0^T\!\!\!…
Jeremy
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"simplifying" an integral to get the desired result

This question is a sequel of this. I tried very hard but i am not able to simplify or beautify this result-$$\sum_{n=0}^\infty (AB)^{n^2}(A/B)^n=\int_{-\infty}^\infty \frac{e^{-t^2}}{\sqrt{\pi}}\frac{1}{1- (A/B)e^{-2t\sqrt{\log AB}}}dt$$ to this…
pjmathematician
  • 542
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How do I show $ \int_{0}^{1} exp(-f(x))dx=\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \int_{0}^{\infty} f(x)^n dx $

I have been given a function f(x) defined on the interval $$[0,\infty] $$ by the formula $$ \begin{array}{cc} \ f(x)= \{ & \begin{array}{cc}\ 0 & x= 0 \\ \ x\space log(x) & x≠0 \end{array} \end{array} $$ I have argumented…
Signe Winkel
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Evaluate $\int_{0}^{1} (x^x)^{(x^x)^{(x^x)^{^{{\cdot}^{\cdot}}}}} \text{d}x$

Evaluate $\int_{0}^{1} (x^x)^{(x^x)^{(x^x)^{^{{\cdot}^{\cdot}}}}} \text{d}x$ This doesn't seem elementary. But since MSE would otherwise want me to show what I have tried, I decided to write it in a more closed form. Write it as $y$, then it becomes…
oldsailorpopoye
  • 520
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What does one integral tell for two variable function?

Let's assume I have function: $ f(x,y) $ Solving double integral: $ \int \int f(x,y)\ dx\ dy $ would give me volume of this body function. For example cubic centimeters. But what if I use only one integral: $ \int f(x,y)\ dy$ What do I get out then?…
filtertips
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Evaluation of $\int_{0}^{\pi/2}\log\left(1+\cos^2 (2x)\right)\, \mathrm{d}x $

$$I=\int_{0}^{\pi/2}\log\left(1+\cos^2 (2x)\right)\, \mathrm{d}x $$ This integral came up while attempting $$I=\int_{0}^{\infty}\frac {\log(1+y^4)}{1+y^2}\, dy$$ Let $x=\arctan(y)$ $$I=\int_{0}^{\frac{π}{2}}(\ln(1+\tan^4 (x))…
Paras
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