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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.

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Integrate negative order square root

I've been trying to solve this integral and can't figure out where to start from: $$ \int\limits _0^1 r\sqrt{\dfrac{4r^2}{(r^2+\varepsilon^2)^2}+1} \textrm{ d}r $$ where $0<\varepsilon<1$.
Gepeto97
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An integral from 0 to infinity

I am trying to show that: $$\int_0^\infty \frac{\sin^{2n-1}x}{x}dx = \int_0^\infty \frac{\sin^{2n}x}{x^2}dx$$ for any positive integer $n$. This eqn struck me when I was evaluating $$\int_0^\infty \frac{\sin^{2n-1}x}{x}dx $$ and $$ \int_0^\infty…
Arpan Mathur
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Definite integral $\int_0^1 \frac{\cos( \pi n x)}{\sqrt{c^2+x^2}} dx$

does anyone have a reference or solution for this integral? $$\int_0^1 \frac{\cos(\pi n x)}{\sqrt{c^2+x^2}} dx$$ $n$ is an integer and $c$ a real number. I tried some tables of integrals and also Mathematica, but I didn't find an answer.
user
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Double integral involving exponential of quadratic form.

Assuming that $a>0$, Maple shows the following $$ \int_0^\infty \int_0^\infty \exp\left(\frac 1 2(-a y^2-2 a y z-a z^2)\right) \, \mathrm dy \, \mathrm dz = \frac 1{a}, $$ whereas $$ \int_0^\infty \int_0^\infty \exp\left(\frac 1 2(-(a+1) y^2-2 a y…
NonalcoholicBeer
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Favorite definite integral trick?

I'm compiling a list of interesting definite integrals for an upcoming blog post, and I thought that the math SE community might have a few interesting problems to offer. I am especially interested in integrals that use "tricks" that are hard to…
Franklin Pezzuti Dyer
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Show that $\frac{-1}{2}<\int_a^b\frac{x^3\cos {5x}}{2+x^2}<\frac{1}{2}$

Let $f(x)\ge g(x)$ for every $x$ in $[a,b]$ and $f$ & $g$ are both bounded and Riemann integrable on [a,b]. At a point $c\in[a,b]$, let $f$ and $g$ be continuous and $f(c)>g(c)$ then prove that $\int_a^b f(x) \ dx>\int_a^b g(x) \ dx$ and hence show…
user467745
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To evaluate the given integral

If $0
Girish Kumar Chandora
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To find the minimum value of given integral.

Consider the polynomial $f(x)=ax^2+bx+c$. If $f(0)=0,f(2)=2$, then find the minimum value of $$\int_{0}^2 |f'(x)| dx.$$ My try: $f(0)=0$ $\implies c=0$ and $f(2)=2 \implies 2a+b=1$ Also $f'(x) =2ax+b\implies f'(1)=2a+b =1\implies f'(1)=1$ Now to…
Girish Kumar Chandora
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To find the number of possible continuous function

What are the number of possible continuous functions $f(x)$ defined on $[0,1]$ for which $$ I_1 = \int_0^1 f(x)dx=1,\\ I_2 =\int_0^1 xf(x)dx =a,\\ I_3 = \int_0^1 x^2f(x)dx=a^2.$$ I have no idea how to to solve it. Can anyone help me?
Girish Kumar Chandora
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Closed-form Expression for Definite Integral of (x-1)/ln(x) dx over a non-negative domain

I understand that integrating (x-1)/ln(x) is a tricky task in the general case, but I am hoping that restricting the problem to a definite integral over a non-negative domain simplifies the answer, to the point that it can be expressed in a closed…
Iron Pillow
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Evaluating the value of $\int_0^1 e^{\left(x^2\right)}~dx+\int_1^e\sqrt{\ln x}~dx$

Evaluate the value of: $$\int_0^1 e^{\left(x^2\right)}~dx+\int_1^e\sqrt{\ln x}~dx$$ Here is what I have tried: $\ln x >\sqrt{\ln x}$ when $x=[1,e]$ $e^x>e^{\left(x^2\right)}$ when $x=[0,1]$ And after that put under the integral and…
Ica Sandu
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Whether the following integral expression can be calculated in closed-form?$\int_0^\infty r \exp(-a_1r^{b_1}-a_2r^{b_2})$

Whether the following integral expression can be calculated in closed-form? $y=\int_0^\infty r \exp(-a_1r^{b_1}-a_2r^{b_2})\mathrm{d}r$, where $a_i,b_i>0,i=1,2$.
Dave
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How to solve this integral using polylogarithms?

Prove that $$I(t) = \int_0^1 \frac{x\ln (1+x)}{1+x^2} \, dx= \frac{\pi^2}{96}+\frac{\ln(2)^2}{8}.$$ I tried this and broke it in a form where I can use 2nd order polylogarithm but could not proceed. This answer is confirmed by Wolfram…
Apollo15
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Integrate arctan from $0$ to $3\pi$

How do I integrate $$\int_0^{3\pi} \frac 1 {\sin^4x + \cos^4 x} \,dx$$ I tried with Weierstrass and obtained:$$ \int\frac 1 {u^2+2}\, du $$ I think it's correct but how do I integrate this given that I cant integrate arctan for $3\pi$ and $0$
Lola
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How to solve integral arctan

I solved this integral: $$ \int_0^{\pi/2}\frac{dx}{5+4\cos x} $$ and I obtained $\frac{2}{3} \arctan \frac{1}{3}$ is it correct? how do I find the answer in $\pi$?
Lola
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