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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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Harmonic number

An integral representation of nth Harmonic number is $$H_n = \displaystyle \int_0^1 \frac{1 - x^n}{1 - x}\,dx$$ Wikipedia states that for every x > 0, integer or not, we have: $$H_{n} = n \displaystyle\sum_{k=1}^\infty \frac{1}{k(n+k)}$$ How can I…
Karlis Olte
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Evaluate $\int_0^1\frac {\{2x^{-1}\}}{1+x}\,\mathrm d x$

I am trying to evaluate $$\int_0^1\frac {\{2x^{-1}\}}{1+x}\,\mathrm d x$$ where $\{x\}$ is the fractional part of $x$. I have tried splitting up the integral but it gets quite complicated and confusing. Is there an easy method for such integrals?
user1488
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Expression for $\int_0^1 x^n(1-x)^{n}/(1+x^2) \ dx$

An answer to this question makes clever use of an integral of this form: $$\int_0^1 \frac{x^n(1-x)^n}{1+x^2} dx$$ Is there a closed form for this for arbitrary positive integer $n$? (I expect this question has been asked before, but I couldn't find…
Simon S
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Defining functions with integrals

Is it okay to have a function $f(x)$ defined by $$f(x)=\int_0^x{x^2-x\,dx}\,?$$ If so, what would $f'(x)$ be? I've seen many questions like these on my math competitions (they give functions defined by integrals and both the limits of the integral…
NestorV S
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How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$?

I need to show that the following integral: $$\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx$$ = $n\pi$, for all natural numbers $n$. What is the method to evaluate the above? I initially thought of the Leibniz rule, but that wouldn't work as the…
Newton
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How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ without changing into polar coordinates?

How to prove $ \int_0^\infty e^{-x^2} \; dx = \frac{\sqrt{\pi}}2$ other than changing into polar coordinates? It is possible to prove it using infinite series?
S L
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An integral of $\,\int_{0}^{\infty}{\!\frac{\ln\left(1+{\frac{1}{x}}\right)}{π^2+\ln^2x}\mathrm{d}x}$

The following is my current solution idea for the integral $$ \displaystyle \begin{aligned}\int_{0}^{\infty}{\dfrac{\ln\left(1+{\large\frac{1}{x}}\right)}{π^2+\ln^2x}\…
Dylan Lee
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Computing $\int_{\Bbb{B}^3\times\Bbb{B}^3}\frac{\rm{d}^3x\,\rm{d}^3y}{\|x-y\|}=\frac{32}{15}\pi^2$, where $\Bbb{B}^3$ is the 3-dimensional unit ball

Let's call $\mathbb{B}^3$ the three-dimensional unit ball, then how to compute the six fold $$ \int_{\mathbb{B}^3\times \mathbb{B}^3}\frac{\text{d}^3x\,\text{d}^3y}{\|x-y\|}=\frac{32}{15}\pi^2 $$ I saw this question on a forum, but no one seemed…
ss12guoji103
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Show that $\int_0^{\pi/2} \frac {\sin(u+a\tan u)} {\sin u}\,\mathrm d u=\pi/2$

I am trying to show that $$\int_0^{\pi/2} \frac {\sin(u+a\tan u)} {\sin u}\,\mathrm d u=\frac {\pi} 2$$ Can this be done via contour integration? I'm not really sure which contour to pick. I have tried substitutions like $\pi/2 - u$ but they haven't…
user544680
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How to find $\int_0^{2\pi} |a\cos(x)+b\sin(x)|dx$, where $a^2+b^2=1$

I need help to solve this integral: $$\int_0^{2\pi} |a\cos(x)+b\sin(x)|dx$$ where $a^2+b^2=1$. I hope someone is able to help me.
Carla Johanson
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What is $\int_0^\infty e^{-e^x}dx$?

Can we compute the following integral? $$\int_0^\infty e^{-e^x}dx\tag{1}$$ I think it converges because $e^x\geq 1+x$ for $x\geq0$, so we have $$0\leq\int_0^\infty e^{-e^x}dx\leq\int_0^\infty…
Vortex
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Finding the integral $\int_{0}^{\infty }\frac{1}{(4x-3)(4x-1)}\,dx$.

Which method that will be effective for solving this integral?
E.H.E
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How do I find the following definite integral?

Find the value of the following definite integral: $$\int_0^{1}\frac{\arctan(t)}{1+t}dt$$ I tried using integration by parts, but it gives another complicated integral, so possibly the antiderivative does not exist. What other methods can I apply?
Shubham
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How to show that $ PV \int_{-\infty}^{\infty} \frac{\tan x}{x}dx = \pi$

In a recent question, it was stated in a comment, without proof, that $$ PV \int_{-\infty}^{\infty} \frac{\tan x}{x}dx = \pi$$ What is the easiest way to prove this? I was able to show that $$ PV \int_{-\infty}^{\infty} \frac{\tan x}{x}dx =…
user111187
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Definite integral problem

I was solving a definite integral problem which was reduced to : $$\int^{1}_{0} \frac{\ln(1+t)}{t} dt$$ I couldn't solve it and when I saw the solution, the answer was simply given as $\frac{\pi^2}{12}$, and claimed that this is an identity. Can…
Cheeku
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