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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

20559 questions
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How to calculate the integral of an inverse function?

If I have this function :$$\ f(x)=x^3+x $$ how can I calculate integral from inverse $$\int_0^1 f^{-1}(x)\,dx $$ ? I understand that in order do solve this problem I have to take $$\ x=f(t) $$ $$\ dx=f'(t) dt $$ and in the end I'll have $$\int…
Lola
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Evaluate $\int_0^{\pi/2}x e^x \sin(x)dx$.

$$I=\int_0^{\pi/2}x e^x \sin(x)dx$$ I gave a lot of efforts in this question and replaced x with $\pi/2 -x $ but to no avail, this even became more complicated. Kindly give me an idea on how to proceed in this very question.
satyatech
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How do we solve integration problem of function of functions of the kind

How do we solve integration problem of function of functions of the kind: $$\int_0^{\pi/2} {\sin (\ln\sqrt {\cos x} )}dx$$ Do we use some substitution e.g., $\sqrt {\cos x} =t$ ? will that work or need another trick?
user12345
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problem on definite integral

I have found in some book the following : $\int^{1}_{0} f(t)t^{n}dt$ = 0 for all $n$ in $N$ iff $f(t)$ = 0 where $f(t)$ is a real valued continuous function on [0,1]. I don't understand the proof of this. How to prove this ?
user21982
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How to calculate integral with n natural number

Prove that $\displaystyle\int_0^1 \dfrac{x^{2n}}{x^{2n}+{(1-x)^{2n}}}dx=\dfrac{1}{2}$ I have only proved that $\displaystyle\int_0^1 \dfrac{x^{2}}{x^{2}+{(1-x)^{2}}}dx=\dfrac{1}{2}$
Nick
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Evaluation of $\int_{0}^{\infty}\frac{x^{a-1}-x^{b-1}}{1-x}dx$

For $0
juantheron
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How to calculate $\int_{0}^{1} \frac{\log{(1+x)}}{x^{2}+1} \ dx $?

$$\int_{0}^{1} \frac{\log{(1+x)}}{x^{2}+1} \ dx $$ I tried substituting x with 1/t but couldn't find the answer. Can someone provide any hint? As many suggested I substituted x with tan t but again I got stuck at $$\int_{0}^{\frac{π}{4}}…
Resorcinol
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definite integral and functions

If $\int_{-\infty}^{\infty}f(x)dx$$=1$ then $\int_{-\infty}^{\infty}f(x-\frac{1}{x})dx$ equals $?$ 1.My try: I tried assuming a function that enclose an area of 1 as it takes values from - $\infty$ to $\infty$ and then replacing $x$ to…
Piyush Raut
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Integral of a gaussian function depending on an angle

Suppose we have the following function: $$\Phi(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{1}{2}\frac{x^2}{\sigma^2})$$ and suppose the variable $x$ is depending on an angle $\alpha$: $$x=sin(\alpha)$$ Is it possible to give an analytic expression…
Riccardo.Alestra
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inequality of definite integrals involving ${F'(x)}^2 $ and ${F(x)}^2$

Please suggest something as I am completely stumped here. $F:[0,\pi] \to \mathbb R$ is smooth function with $F(0)=0=F(\pi)$.We need to show that: $$ \int_0^\pi {(F'(t))}^2dt \ge \int_0^\pi {(F(t))}^2 dt$$ Thanks,any hint also would be appreciated.
TheJoker
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What is the integral of $e^{\delta(x-x_0)}$?

Simple question, what is the answer to the following integral: $\int_{-\infty}^\infty \exp[\delta(x-x_0)] dx$ Is it 1? Furthermore, is there an integral to a similar function, say, $\int_{-\infty}^\infty \exp[\delta(x-x_0)]\exp[i(x-x_0)^2]dx$
Lou
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Value of $a$ for which two integrals are equal

Determine $a$ such that $$\int _0 ^a ([\arctan \sqrt {x}]dx=\int _0 ^a [\frac {\pi}{2}-\arctan\sqrt {x}]dx$$ where $[.] $ is greatest integer function so i first took the first integral to get integral as $a [\arctan (a)] $ as I think derivative…
Archis Welankar
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Definite integral FTC

Find a function f so that: $\int _a^{x^2}\:f\left(t\right)ln\left(t\right)dt\:=\:x^3\left(ln\left(x\right)-\frac{1}{3}\right)$, $a>1$ This is how I did it: First get the derivative of $x^3\left(ln\left(x\right)-\frac{1}{3}\right)$ which is…
bajotupie
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double integral wrt minimum

How can one calculate the following integral? $$ \int_0^1{\int_0^1 xy \quad d[min(x,y)]} $$ I have no idea how to hande the $d(min(x,y))$. Does anyone have an idea for this problem?
DeMerlo
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Evaluation of $\int_\limits{-1}^{3}\frac{x\ln |x|}{1+x^4}dx$

Evaluation of $$\int_\limits{-1}^{3}\frac{\arctan(1+x^2)}{x}dx$$ $\bf{My\; Try::}$ Let $$I = \int_\limits{-1}^3\arctan(1+x^2)\cdot \frac{1}{x}dx$$ Using By parts, We get $$I = \left[\arctan (1+x^2)\cdot…
juantheron
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