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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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Stuck on contour integral

$$\int_0^\infty \dfrac{\sin x}x\,\mathrm dx = \mathrm{Im}\left(\int \dfrac{e^{ix}}x\,\mathrm dx\right)$$ Considering $f(z) = \dfrac{e^{iz}}z$, it has a simple pole at $z = 0$. Residue of $f(z)$ at $z = 0$ is $1$. This imples…
ENEJO SUNDAY OCHE
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Evaluate $\int_{0}^{l-a} \frac{dx}{\sqrt{(a+x)^2-a^2}} $

Willing to evaluate $$\int_{0}^{l-a} \frac{dx}{\sqrt{(a+x)^2-a^2} }$$ The answer is given $ \cosh ^{-1} (l/a)$. Don't know how this can be achieved. Please help
user533926
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Let $I=\int_{-\pi}^{\pi}\dfrac{sin^2x}{1+a^x}dx,a>0$.Find the value of $I$

I have taken $a=e$ but using partial integration, getting no luck.Please help.I avoided purposefully 6 pages of needless scribble as that may deviate the answerer.
Saradamani
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Does the following integral converge, and if yes, to what?

Can someone help me solve the following definite integral? $$\mathcal I(a) = \int_{-\infty}^\infty \frac{\mathrm{e}^{-i |t| \sqrt{a^2+x^2}}} {\sqrt{a^2+x^2}}\,dx\,. $$ This can be solved, I believe, by analytic continuation to complex numbers and…
Nanashi No Gombe
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Help for calculating Area Using Double Integral

**I am trying to solve this problem. I would appreciate if someone can check if my approach is correct. Problem: D is the plane region defined by: $$x^2+y^2-2x \le0\ and\ y\le |x| $$ $a)\ Calculate\ the\ Area\ D$ $(b)\ Evaluate$ $$\iint_D…
Soso
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An interesting relation of integrals

Proof that there does not exist an $F: \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ such that $\int_{0}^{1} f(x)g(x)dx =F(\int_{0}^{1}f(x)dx, \int_{0}^{1}g(x)dx).$
Valy
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Cylindrical shells word problem. Did I set this up correctly?

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. $xy = 1, x = 0, y = 1, y = 3$, rotated about the x-axis. So I plan to do this via integration using…
Jwan622
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Double integral over an area

Find the double integral of $x$ over $S$ where $S$ is the area bounded by $x^2 + y^2 \leqslant 2$ and $x \geqslant 1$. Use the following subtitution: $x=r\cosθ ,y=r\sinθ$. I have separated the original integral (let it be $I$) into $I=J-K-2L$,…
GiaFil7
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Definite integral problem.

I am stuck on this one: $$ \int_0^{\frac{\pi}{4}} \frac{1 - \cos^2{\theta}}{\cos^{2}{\theta}} d \theta$$ $$ = \int_0^{\frac{\pi}{4}} \tan^{2} \theta$$ What is the antiderivative of $\tan^2{\theta}$ Is it this: Since: $$\frac{d}{dt} \tan{t} =…
Jwan622
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Definite integral. My answer is different from Wolfram alphas. Why?

I am trying to solve this definite integral: $$\int_0^1 ( x^{10} + 10^{x} )$$ $$ \left[ \frac{x^{11}}{11} + \frac{10^x}{\ln 10} \right ]_0^1$$ $$ \frac{1}{11} + \frac{10}{\ln{10}} - 1$$ But wolfram alpha says this: How do I unite the two?
Jwan622
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Definite integral notation correct?

I'm having trouble understanding some of the notation for definite integrals. Is this right? $$ \int_1^2 (\frac{1}{x^2} - \frac{4}{x^3}) \, dx$$ $$ \int_1^2 x^{-2} - 4x^{-3} \, dx$$ $$ = \int_1^2 x^{-2} \, dx- \int_1^2 4x^{-3} \, dx$$ $$ \left[…
Jwan622
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Evaluate this integral. I think I'm missing something.

$$\int_1^{18} \sqrt{\frac{3}{z}}dz $$ So I have to find the antiderivative of $\sqrt{\frac{3}{z}}dz$ = $\frac{2}{3} \cdot \frac{3}{z}^{\frac{3}{2}}$ So the integral is: $$\left[ \frac{2}{3} \cdot \frac{3}{z}^{\frac{3}{2}} \right ]_1^{18}$$ But I am…
Jwan622
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Finding volume bound by unsymmetrical paraboloid

So, going through triple integral volume problems, when establishing the bounds of integration for a solid bound by a paraboloid, one common trick is to convert to cylindrical coordinates allowing for the substitution of x^2+y^2 to r^2. However,…
Math Loser
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Mistake in definite integral of $\int_{-1}^{1} d/dx \arctan(1/x) dx$

$$\int_{-1}^{1} \left(\frac{\mathrm{d}}{\mathrm{d}x}\arctan{\frac1x}\right) \, \mathrm{d}x$$ Since integration cancels the differentiation, we need $\arctan{(1)} - \arctan{(-1)}$, that is $\pi/2$ But in the book the answer is $-\pi/2$ , so I think…
Reginold
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Integration of rational function in terms of $p$ and $q$

Evaluation of $\displaystyle \int^{\infty}_{0}\frac{x^{p-1}}{1+x^q}dx$ My attempt: put $\displaystyle1+x^q=\frac{1}{t}$. Then $qx^{q-1}dx=-\frac{1}{t^2}dt$. So integral $$I=\frac{1}{q}\int^{1}_{0}\frac{(1-t)^{\frac{p-q}{q}}}{t^{p-q}}dx$$ It seems…
DXT
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