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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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$M = \int_0^{\pi/2} \frac{\cos x}{(x+2)} dx$ and $N = \int_0^{\pi/4} \frac{\sin x \cos x}{(x+1)^2} dx$, then value of $M - N$ is?

Options are : (A) $\pi\quad$ (B) $\frac{\pi}{4}\quad$ (C) $\frac{2}{\pi + 4}\quad$ (D) $\frac{2}{\pi - 4}$ I have solved it using Taylor's expansion of the numerators. E.g, $\cos x = \cos ((x+2) -2) = \cos(x+2) \cos 2 + \sin(x+2) \sin 2$,…
Monalisha Bhowmik
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Evaluate: $\int_0^1 \frac{e^x(1+x) \sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx $

Evaluate: $\int_0^1 \frac{e^x(1+x)\sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx $ My assumption put $t= x e^x $ then $\int_0^e \frac{\sin^2(t)}{\sin^2(t)+ \sin^2(e-t)} \,dt$ How can I proceed from here? Thanks in advance.
Chris
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Evaluation of double integration with polar coordinates

Evaluation of $$\iint_R e^{-(x^2+y^2)} \, dA$$ where $R$ is the region given by inequalities $x^2+y^2\leq 1$ and $0\leq y\leq\sqrt{3}x$ What i try:: drawing Circle and line. We get a sector which makes an angle of $\pi/3$ with $x$ axis. Now put…
jacky
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How to find an unknown exponent given the area between 2 functions?

Let $a > 0$ and consider the functions $f(x)=xe^{ax}$ and $g(x) = x$ a) Determine the value of a, given that the area between these functions for $0≤x≤1$ is equal to $1$ square unit. My attempt: I know that for x > 0, f(x) > g(x) so I am going to…
CountDOOKU
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Find the area of the finite region between AB and the curve.

The diagram shows part of a sketch of the curve with equation $y=\frac{2}{x^2}+x$. The points $A$ and $B$ have $x$-coordinates $\frac 12$ and $2$ respectively. Find the area of the finite region between $AB$ and the curve. First, I found the $y$…
MrRookie
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Definite integral involving square roots

Is there a simple way to compute the following integral $$ I(a)\equiv \int_0 ^{+\infty}\left[\frac{2 a^{3/2} \left(x^2+1\right)^{3/2}}{\sqrt{a \left(x^2+2\right)+1}}-a \left(2 x^2+1\right)+1\right]\mathrm{d}x $$ with $a>0$. Using Mathematica, I…
user12588
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The integral: $\int_{0}^{\pi/2} \frac{\sin x \cos^5 x}{(1-2\sin^2x\cos^2x)^2}dx$

The integral: $$\int_{0}^{\pi/2} \frac{\sin x \cos^5 x}{(1-2\sin^2x\cos^2x)^2}dx$$ has been encountered today while solving a longlish problem at MSE. The question here is: How would one evaluate it? Addendum For an interesting use of this integral…
Z Ahmed
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Integral $\lim_{\epsilon\to 0^+} \int_{\Lambda/s}^\Lambda \int_{-\infty}^\infty \frac{i}{y^2-x^2+i\epsilon} dydx = \pi\log s$

I am stucked with the following integral $$\lim_{\epsilon\to 0^+} \int_{\Lambda/s}^\Lambda \int_{-\infty}^\infty \frac{i}{y^2-x^2+i\epsilon} dydx = \pi\log s.$$ Here $s,\Lambda$ are positive constants. This integral is motivated from physics. How…
Laplacian
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Volume of Cylinder cut by a plane

I am trying to solve this problem: $$2\int_{-2}^2\int_0^{\sqrt{4-x^2}}(4-y)\,dx\,dy$$ This is supposed to be the area of a cylinder defined by $x^2+y^2=4$ and cut by the plane $z=4-y$. Is this even the proper integral to use? The bounds on the…
user753407
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definite integration with infinite limit

If $a>0$ and $b>1$ and $f:\left[0,1\right] \rightarrow \mathbb{R}$. Then value of $\displaystyle \lim_{n\rightarrow \infty}n^{\frac{a}{b}}\int^{1}_{0}\frac{f(x)}{1+n^{a}x^{b}}dx$ is What i try For $0\leq a<1$ $$I =\lim_{n\rightarrow…
jacky
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Definite Integral of $(a^2-x^2)^\frac{3}{2}$

Prove the following: $$\frac{4c}3\int\limits_0^a\left(a^2 - x^2\right)^\frac32\,\mathrm dx = \frac{\pi a^4c}4.$$ Taking $x = a\sin\theta$, how will the limit change?
danny
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Integrate the GIF

Solve $$\int_0^1 f(x)= \left[\frac1{3x}\right] - \frac13\left[\frac1x\right]\text dx$$ where $[x]$ represents greatest integer less than or equal to $x$. My attempt I substituted $3x$ with $t$, to get $$\int_0^1\frac13\left[\frac1x\right]$$ from…
Som
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The definite integral of $e^{-x^2/2}$ on the interval $[-a/2,a/2]$, Is there any explicit solution?

The definite integral of $e^{-x^2/2}$ on the interval $[-a/2,a/2]$, Is there any explicit solution?
张Fous
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Integral of a log reciprocal

I want to calculate following definite integral: $$\int\limits_{-1}^0\left(\frac{1}{x\ln(1 + x)}-\frac{1}{x^2}-\frac{1}{2x}\right)dx$$ No idea from where to begin. I know result is: $$-\frac{1}{2}-\frac{\gamma_0}{2}+\frac{\ln(2\pi)}{2}$$
24th_moonshine
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Definite integral help

I'm working on a physics problem and I got to the integral: $$\int_0^\infty (a+b+x^2)^{-\frac{3}2} dx = \frac{1}{(a+b)}$$ I am just trying to understand how this is achieved. Because the indefinite integral…
abyssmu
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