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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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How to evaluate this double integral over a semicircle?

$$\iint_{D}\frac{x^4+y^4}{1+e^{3x^2y-y^3}} dxdy$$ $D = \{(x,y):x^2+y^2 \leq 1,x>0\}$. From the shape of region $D$ it seems to me that it's better to convert it to polar coordinate,but with no luck.
Metta World Peace
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Integral $ \int_{-\infty}^\infty \frac{1}{x^2+1} \left( \tan^{-1} x + \tan^{-1}(a-x) \right) dx$

For $a\in\mathbb R$, I want to evaluate the integral $$ I = \int_{-\infty}^\infty \frac{1}{x^2+1} \left( \tan^{-1} x + \tan^{-1}(a-x) \right) dx.$$ I tried to integration by parts by considering $\left( \tan^{-1}(x) \right)' = \frac{1}{x^2+1}$, so…
Laplacian
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Integral$ \int_0^{\pi\over3} \frac{\cos^2x}{\sqrt{1+\cos^2x}} dx$

$$\int_0^{\pi\over3} \frac{\cos^2x}{\sqrt{1+\cos^2x}} dx$$ I tried to substitute $1+\cos^2x$ , tried to change cos in sin by complementary formula etc . But nothing seems to work out. I think it's not as simple as I thought ?
RKK
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A definite integration problem

$$\int_{-\pi}^\pi \frac{\cos^2x}{1+a^x}dx$$ where $a>0$ This question is from my textbook. I am finding this question quite difficult to solve.
Krish
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an integral from the tables

The following integral occurs in the book, “Integrals and Series“ v. I , Prudnikov et all, page 542: $\displaystyle \int_{0}^{\infty}{\ln|\cos(ax)|\frac{1}{x^2+z^2}dx}=\frac{\pi}{2z}\ln{\frac{1+e^{-2az}}{2}}$ I have had no luck in verifying this…
Jet Foncannon
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Prove or disprove that $\int_{-1}^{1}\left(x+\frac1x\right)\cos x\,dx=0$

Prove or disprove that $\int_{-1}^{1}\left(x+\frac1x\right)\cos x\, dx=0$ My working: If $f(x)$ is an odd function, then $\int_{-a}^{a}f(x)dx=0$ As $\left(x+\frac1x\right)\cos x$ is odd, $\int_{-1}^{1}\left(x+\frac1x\right)\cos x\, dx=0$ Is this…
Makar
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An integral about trigonometric function.

Recently I met an integral which is $\int_0^\infty \left(\frac{\sin x}{x}\right)^3 \; dx$. I get the result is $3\pi/8$ by using Mathematica, but I cannot derive it independently. So I hope someone can help me. It is my first time to ask questions…
dezhonger
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Setting up proper integral

I can't get a problem to work out the way it is supposed to according to my book. First, a relevant theorem for the problem: THEOREM Suppose $\{V_j; j \in Z\}$ is a multiresolution analysis with scaling function $\phi$. Then the following scaling…
Kristian
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Approximation of Arc Length Curve

I was studying how to calculate arc length from this page: https://tutorial.math.lamar.edu/classes/calcii/arclength.aspx And it said that Why this statement $L= \lim\limits_{n \to \infty}\sum_{i=1}^{n}|P_{i-1}P_i|$ is true? Why is it exact and not…
Elsa Hejazian
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Definite integration with unknown function f

I came across this question in a practice test, but I have no idea where to even begin. If a function $f$ is continuous on $[0, 1]$, show that $$\lim_{n\rightarrow \infty}\int_0^1 \frac{nf(x)}{1+n^2x^2} dx = \frac{\pi}{2}f(0)$$ Any hints would be…
utkarshk5
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Integral involving gaussian function

I would like to calculate the following integral: $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\quad (x^2+y^2)^k\exp\left(-\dfrac{(x-x_0)^2+(y-y_0)^2}{a^2}\right)\,\mathrm dx\,\mathrm dy$$ Any clue on how to proceed? Thanks
JFNJr
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Bounding integrals using Maclaurin series

Prove that $$\int_0^1 \frac{ \sin x}{\sqrt{x} } < \frac23$$ I didn't see any easy way to integrate directly, so I just used the series approximation: $$ \sin(x) = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!}$$ Hence, $$\int_0^1 \frac{ \sin…
tryst with freedom
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By using the properties of definite integrals, evaluate $\int_0^{\pi}\frac{x}{1+\sin x}dx$

By using the properties of definite integrals, evaluate $\int_0^{\pi}\frac{x}{1+\sin x}dx$. My attempt: (Using the property $\int_0^{2a}f(x)dx=\int_0^a(f(x)+f(2a-x))dx$) $$\int_0^{2\frac{\pi}{2}}\frac{x}{1+\sin…
aarbee
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Is this Riemann sum equivalent to this definite integral?

For the following scenario, The density of people in a 200-ft long stadium during a concert is given by c(x), where x is the distance, in feet, from the stage. Find the number of people at the concert who are at most x feet away from the…
Tom
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how can i solve the following integral

$$I = \int _{a}^{a+h} x^n \alpha h^{-\alpha}(x-a)^{\alpha-1} dx$$ $$I = \alpha h^{-\alpha} \int_{a}^{a+h}x^n (x-a)^{\alpha-1} dx$$ The result will be $$\sum_{k=0}^{n} {n \choose k} h^k a^{n-k} \frac{\alpha}{\alpha+k}$$
time
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