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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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Evaluation of $\int^{\infty}_{0}(2x+1)e^{-x^{3}}dx$

Evaluation of $$\int^{\infty}_{0}(2x+1)e^{-x^{3}}dx$$ What i try:: Let $x^{\frac{3}{2}}=t$. Then $\displaystyle \sqrt{x}dx=\frac{2}{3}dt$ and changing the …
jacky
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$\int_{-\pi}^{\pi} \frac{2x(1+\sin(x))}{1+\cos^2(x)} \mathrm d x$

$$\int_{-\pi}^{\pi} \frac{2x(1+\sin(x))}{1+\cos^2(x)} \ \mathrm dx$$ My attempt: $$\int_{-\pi}^{\pi} \frac{2x(1+\sin(x))}{1+\cos^2(x)} \ \mathrm dx =\int_{-\pi}^{\pi} \frac{2x}{1+\cos^2(x)} \ \mathrm dx+\int_{-\pi}^{\pi}…
DatBoi
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Integral $\int_{-D}^D \frac{1}{x-ia} \log \frac{(x-2D)^2+a^2}{x^2+a^2} dx$

For $a>0$ and $D>0$, I want to evaluate the integral $$\int_{-D}^D \frac{1}{x-ia} \log \frac{(x-2D)^2+a^2}{x^2+a^2} dx.$$ Here $i$ is the imaginary unit. This integral arises when evaluating a Feynman diagram in physics.
Laplacian
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Integral $ \int_{-\infty}^{+\infty} \frac{ \sin^2(\sqrt{(x-a)^2 + b^2}\,\,t)}{(x-a)^2 + b^2} dx$

I am trying to solve the following definite integral: $$ \int_{-\infty}^{+\infty} \frac{ \sin^2(\sqrt{(x-a)^2 + b^2}\,\,t)}{(x-a)^2 + b^2} dx$$ with $a$ and $b$ being real constants. Notice that the integrand is non-negative for all x , with a peak…
Karthik
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Finding $(a,b)$ that minimizes $I=\int_a^b(x^4−2x^2)\,\mathrm dx$

Let $\displaystyle I=\int_a^b(x^4−2x^2)\,\mathrm dx$, then $I$ reaches the minimum when the ordered pair $(a,b)$ is:$$(-\sqrt2,0)\quad(0,\sqrt2)\quad(\sqrt2,-\sqrt2)\quad( -\sqrt2, \sqrt2)$$ I solved the integration and got…
aarbee
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Substitution in Definite Integration

$$\int_0^{\pi} \frac{(\sec x)^2}{2+(\tan x)^2} dx $$ I have made the substitution of of $\tan x=u$ but that makes the lower limit and upper limit both equal to $0$ since $\tan \pi=0$ and my result becomes $0$. What mistake am I making here?
Anurag Saha
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Evaluate the integral $\int\limits_0^{\pi}\log (1-\cos x)\, dx$

Question: What is the value of the integral $\int\limits_0^{\pi}\log (1-\cos x)\, dx$ Proceed: Let $I=\int\limits_0^{\pi}\log (1-\cos x)\, dx$, then $I=\pi \log 2+\int\limits_0^{\pi}\log \sin^2(x/2)\, dx$. What can I do now?
MKS
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For continuous $g$, prove that $1+\left(\int^{1}_{0}g^2(x)dx\right)^3>\left(\int^{1}_{0}g(x)\right)^3$

If $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function Then Prove that $$1+\bigg(\int^{1}_{0}g^2(x)dx\bigg)^3>\bigg(\int^{1}_{0}g(x)dx\bigg)^3$$ My attempt: $\bullet\;\; $If $|g(x)|\geq 1,$ Then $\displaystyle \int^{1}_{0}g^2(x)dx\geq…
jacky
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Solution for the integral expression that is equivalent to pi.

I am trying to evaluate the following integral: $$ \int_{-2}^2 \left(x^3 \cos \left(\frac{x}{2}\right)+\frac{1}{2}\right) \sqrt{4-x^2} \, dx. $$ The answer is pi based on online calculations. But I want to share this to my students that we can…
PRD
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Swapping the limits of integral should change the sign?

I have an elementary question. In this video(Link), Sal explains why swapping the limits of integration changes the sign of the result; In his video, he reasons that "dx"s in the reverse direction must have opposite sign too (because (a-b)/n must…
aderchox
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Computing double integral $\ \iint \sqrt{4-r^2}r \ dr d\theta $

I am trying to solve the following integral: $$\ 2 \int_0^\pi \int_0^{2\sin\theta} \sqrt{4-r^2} \ r \ dr d\theta $$ My attempt: $$\ 2 \int_0^\pi \int_0^{2\sin\theta} \sqrt{4-r^2} \ r \ dr \ d\theta \stackrel{t = 4-r^2}{=} -\int_0^\pi \int…
bm1125
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The region bounded by $y^2=4ax$, the $x$-axis, and the line $x=a$ is rotated about $x=-a$. Find the volume of the solid.

The plane figure bounded by the parabola $y^2=4ax$, the x-axis and the ordinate at $x=a$, is rotated a complete revolution about the line $x=-a$. Find the volume of the solid generated. I tried to integrate this by putting a and - a as the upper…
Orestes Dante
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$\int_0^\infty f(x) dx =a, 0\frac{a^2}{2}$.

Suppose $f(x)$ is continuous on $[0,+\infty)$. Show that $\int_0^\infty f(x)dx=a, 0\frac{a^2}{2}$. My try: $\int_0^\infty (x-a/2)f(x)dx\geq \int_t^\infty (x-a/2)f(x)dx...$ how to use $0
xldd
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Evaluate $\int_{0}^{\pi}x\ln[b^{2a}+2b^a\cos(ax)+1]dx$

How to evaluate this integral? Let $(a,b)\ge2$ $$I=\int_{0}^{\pi}x\ln[b^{2a}+2b^a\cos(ax)+1]\mathrm dx$$ $u=\ln[b^{2a}+2b^a\cos(ax)+1]$ $u^{'}=\frac{2ab^a\sin(ax)}{b^{2a}+2b^a\cos(ax)+1}$ $v=\int x=\frac{x^2}{2}$ $$I=\frac{x^2}{2}\cdot…
user569129
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3 answers

How to prove $\int^{\pi}_0 \cos(x)\log(\tan(\frac{x}{4}))\,\mathrm dx = -2$

How to prove $\int^{\pi}_0 \cos(x)\log(\tan(\frac{x}{4}))\,\mathrm dx = -2$? Or $\int^1_0 \log(u)\frac{(u^4-6u^2+1)}{(u^2+1)^3} \,\mathrm du = -1/2$. (I substituted $\tan(x/4) = u$.)
Kimuppupp
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