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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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Is this a valid operation or an abuse of notation?

So I was thinking about the formula for integrals: $\int f(x) dx$. But since $y = f(x)$, I started to wonder about $y dx$ and using a parametrisation of $x$ and $y$ in the integral formula to solve definite integrals. For instance, I considered the…
Cataline
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Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } $$ One solution: Consider : $\displaystyle F(m,n)=\int _{ 0 }^{ \pi /2 }{ \sin ^{ 2m-1 }{ x } \cos ^{ 2n-1 }{ x } dx } $ To solve this put $\sin^{2}x = t $ to get our…
User1234
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positive double integrals

Suppose $U:={\iint}_{R} (x^2 + 2y^2+9) \,dx\,dy$ and $V:= \iint _R (2x^2 + 3y^2)\, dx\,dy$. Determine the integration region $R$ where $U \geq V$. Hence, find the value $U-V$ over this region. Attempt: Since $U \geq V$, we have $\iint _R(-x^2 - y^2…
Idonknow
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What is $\int_0^1 \frac{\log(x+1)}{x^2+1}dx$

It's so deceptively simple and none of the usual techniques are working. Any and all insights are welcome.
Archaick
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Can this integral be expressed in terms of elementary functions?

Let $\gamma>1$ be fixed. For $1\leq x\leq y\leq \gamma$, I would like to evaluate the following integral : $$F(x,y)=\frac{1}{\pi}\int_1^x\frac{dt}{t\sqrt{t-1}}\arcsin \sqrt{\frac{y-t}{\gamma-t}}.$$ What I know about this are the boundary values : 1)…
MassiveJack
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Why do we set $x+1=\frac{1}{t}$ When we compute $I= \int_{{1}/{2}}^{0} \frac{dx}{(x+1)\sqrt{(3-x)(x+1)}}.$

When we compute $I=\displaystyle \int\limits_{\frac{1}{2}}^{0} \dfrac{dx}{(x+1)\sqrt{(3-x)(x+1)}}.$ We set $x+1=\dfrac{1}{t}$ and we have $\displaystyle I=\int\limits_1^2\dfrac{dt}{\sqrt{4t-1}}$. I have a question : "Why do we set…
Hung nguyen
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Integrating inverse trigonometric function over square root, definite integral.

I haven't been able to figure out how to do this definite integral, and I know it has a known answer. Can anyone help me work out the details for this? The answer (and solution) are given by $$\int_u^1\frac{\cos^{-1}x\,\mathrm…
user2730279
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Area under region $y=log(x)$

I encountered a problem that asks me to calculate the area between the curves $y=0$, $y=-2$, $y=log(x)$, and $x=0$. But in order to do so, it requires to calculate and use the following integral: $\int_{-2}^{0}{e^x}{dx} = 1 - \frac{1}{e^2}$. I can't…
Johnny
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$ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$

What are some different methods to evaluate $$ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$$ for $a > 0$. This integral arises in a number of contexts in Physics and was the original motivation for my asking. It also arises naturally in statistics as…
Simon S
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Integral $\int_{0}^{\infty}\frac1{\sqrt[\alpha]{1+x^\beta}}dx$

Is there a general answer for the integral of the form:$$\int_{0}^{\infty}\dfrac1{\sqrt[\alpha]{1+x^\beta}}dx$$
jimjim
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How to proceed with the following integration?

If $n$ is a positive integer, show that $$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2) dt = \frac{(-1)^n}{c}$$ for some $c \in [\sqrt{n\pi}, \sqrt{(n+1)\pi}]$ I have an idea that i can use Mean value Theorem, but I am not able to proceed. Please…
Manish
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Volume of a split log

When solving the following problem, I could not understand why my reasoning came up with an answer that's different than the one on the solution's manual. Question: Consider $(x,y,z)$ such that $x^2+y^2<1, x>0, 0 \le z \le 5$. This describes one…
MBdr
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How find integral $\int_{[0,\frac{\pi}{4}]^n}\frac{dx_{1}dx_{2}\cdots dx_{n}}{\sqrt{1+\sec^2{x_{1}}+\sec^2{x_{2}}+\cdots+\sec^2{x_{n}}}}$

show that $$\int_{[0,\dfrac{\pi}{4}]^n}\dfrac{dx_{1}dx_{2}\cdots dx_{n}}{\sqrt{1+\sec^2{x_{1}}+\sec^2{x_{2}}+\cdots+\sec^2{x_{n}}}}=\dfrac{\pi^n(n!)^2}{(2n+1)!}$$ first I see this problem I want use $\sec{x_{i}}=y_{i}$,But follow is very…
math110
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Evaluating definite integral

I'm having a hard time understanding how to compute this integral. $$\int_1^4\frac{3x^3-2x^2+4}{x^2}\,\mathrm dx$$ The steps I do is $\dfrac{3x^4}{4} - \dfrac{2x^3}{3} + 4x$ but I don't know how to integrate the $x^2$ in the integral. I know it's…
CMLara
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Integrating $\int_0^{\pi/2} {\sin^2x \over 1 + \sin x\cos x}dx$

We need to evaluate $\displaystyle \int_0^{\pi/2} {\sin^2x \over 1 + \sin x\cos x}dx$ and some solution to this starts as, $\displaystyle\int_0^{\pi/2} {\sin^2x \over 1 + \sin x\cos x}dx = \int_0^{\pi/2} {\{\sin(\pi/2 -x)\}^2 \over 1 + \sin (\pi/2…
Masroor
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