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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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Can we simplify $\int_0^{\pi}\left(\frac{\sin nx}{\sin x}\right)^mdx$?

Letting $m,n$ be natural numbers, can we simplify the following definite integral? $$\int_0^{\pi}\left(\frac{\sin nx}{\sin x}\right)^mdx$$ I've known the followings: $$\int_0^{\pi}\frac{\sin 2kx}{\sin x}dx=0, \int_0^{\pi}\frac{\sin (2k-1)x}{\sin…
mathlove
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Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$.

This question was asked in an entrance test for an undergraduate mathematics program today, held all over India. Question: $f$ is a differentiable function in $[0,1]$ such that $f(f(x))=x$ and $f(0)=1$. Find the value of $\int_{0}^{1}…
Potemkin Metro Card
  • 1,598
17
votes
1 answer

Finding the reason behind the value of the integral.

I was just trying to find $$\int_{0}^{\pi / 2}\frac{\sin{9x}}{\sin{x}}\,dx $$ using an online integral calculator. And surprisingly I found that if I replace $9x$ by $ x,3x,5x$ which are some odd multiples of $x$ the value of integral came out to be…
Jasmine
  • 598
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2 answers

What is $\int_{0}^{\pi}\frac{x}{x^2+\ln^2(2\sin x)}\:\mathrm{d}x$?

Could you prove that: \begin{align} \displaystyle 2\left(\int_{0}^{\pi}\frac{x}{x^2+\ln^2(2\sin x)}\:\mathrm{d}x\right)^{7} + & 53 \left(\int_{0}^{\pi}\frac{x}{x^2+\ln^2(2\sin x)}\:\mathrm{d}x\right)^{5} \\ +…
Olivier Oloa
  • 120,989
13
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2 answers

Find $\int_{0}^{\frac{\pi}{2}} \ln(\sin(x)) \ln( \cos(x))\,\mathrm dx$

$$\large\int \limits_{0}^{\frac{\pi}{2}} \ln(\sin(x)) \ln( \cos(x))\mathrm dx$$ TL;DR: Is there an elegant way of integrating this? I've reduced it to a series, detailed below, but the closed form eludes me, and the only solution I've seen uses a…
Meow
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13
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2 answers

Bound on integral

I need lower and upper bounds (as tight as possible) on the following integral: $$\int_0^\infty x^n\exp\left(-\frac{(x-x_0)^2}2\right)\, dx$$ $n$ is a real number greater than $0$, and $x_0>0$. I am guessing the bounds will be of the form…
Ivan
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12
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Evaluate $\int_0^{\pi/4} \frac {\sin x} {x \cos^2 x} \mathrm d x$

I'm trying to evaluate: $$\int_0^{\pi/ 4} \frac {\sin x} {x \cos^2 x} \mathrm d x$$ Mathematica gives the numerical approximation: $0.959926156626593638859649248036004150970933774605514278777212260466184427508$ I cannot find a closed form though.…
user85798
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4 answers

Definite Integral $\int_0^1\frac{\ln(x^2-x+1)}{x^2-x}\,\mathrm{d}x$

$$\int_0^1\frac{\ln(x^2-x+1)}{x^2-x}\,\mathrm{d}x$$ WA gives $\pi^2/9$
user85798
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2 answers

Reduction formula for integral $\sin^m x \cos^n x$ with limits $0$ to $\pi/2$

I found in this link the reduction formula $$ \int_0^{\pi/2} \sin^m x \cos^n x \ dx = \begin{cases} \frac{[(m-1)(m-3) \cdots 1][(n-1)(n-3) \cdots 1]}{(m+n)(m+n-2) \cdots 2} (\frac{\pi}{2}) & m, n \text{ even}\\[5pt] \frac{[(m-1)(m-3) \cdots (2\text{…
Astrid A. Olave H.
  • 515
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3 answers

Evaluate $\int_0^1{\frac{y}{\sqrt{y(1-y)}}dy}$

I'm a little rusty with my integrals, how may I evaluate the following: $$ \int_0^1{\frac{y}{\sqrt{y(1-y)}}dy} $$ I've tried: $$ \int_0^1{\frac{y}{\sqrt{y(1-y)}}dy} = \int_0^1{\sqrt{\frac{y}{1-y}} dy} $$ Make the substitution z = 1-y $$ =…
Amos Joshua
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Is there an analytical solution to $\int_1^\infty \frac {dx}{\prod_{i=0}^n (x+i)}$

In a problem from physics, I have to deal with this apparently simple function $$I_n=n! \int_1^\infty \frac {dx}{\prod_{i=0}^n (x+i)}$$ $(n\geq 1)$, the result of which being, for sure, something like $\log\left(\frac {p_n}{q_n}\right)$ where…
Claude Leibovici
  • 260,315
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1 answer

How to prove the second mean value theorem for definite integrals

It's a variant form of the second mean value theorem. [Theorem] If $f$ is integrable on [a, b], (i) if $g$ is monotonically decreasing on [a, b], and $g(x)\ge0$, then there exists $e\in[a,b]$, that$$ \int_a^bf(x)g(x)dx=g(a)\int_a^ef(x)dx $$(ii) if…
Winter Young
  • 297
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1 answer

Evaluate $ \int_{0}^{1}{\frac{\ln{x}}{x^2-x-1}dx}$

How can we evaluate the definite integral $$\displaystyle \int_{0}^{1}{\frac{\ln{x}}{x^2-x-1}dx}$$ I tried many times but still have no idea.
pxchg1200
  • 2,050
11
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5 answers

$\int_{0}^{1}\frac{\sin^{-1}\sqrt x}{x^2-x+1}dx$

$\int_{0}^{1}\frac{\sin^{-1}\sqrt x}{x^2-x+1}dx$ Put $x=\sin^2\theta,dx=2\sin \theta \cos \theta d\theta$ $\int_{0}^{\pi/2}\frac{\theta.2\sin \theta \cos \theta d\theta}{\sin^4\theta-\sin^2\theta+1}$ but this seems not integrable.Is this a wrong…
user1442
  • 1,212
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1 answer

Evaluate the integral $\int_{-\pi/2}^{\pi/2}\frac{1}{1+2009^x}\frac{\sin(2010x)}{\sin(2010x)+\cos(2010x)}\,\mathrm{d}x $

Any hints for this one please? $$\int_{-\pi/2}^{\pi/2}\frac{1}{1+2009^x}\frac{\sin(2010x)}{\sin(2010x)+\cos(2010x)}\,\mathrm{d}x $$
Superbus
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