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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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Double Integral for volume of tilted ellipse solid

I am trying to determine the volume of the blue shaded area in the first figure. This volume is essentially the volume between two tilted ellipses (one up and one down). Since I am not well versed with double integrals, I will include comments on…
rdemo
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What is the analytic solution to the definite double integral of x^i dxdi over some finite positive domain?

Ok so I am very confused about the answer to this integral that I am getting. For some positive finite domain $A$ I want to solve $$ \iint_{A} x^y dxdy $$ When I throw this in desmos' 3d plotter I have no issues and get finite values, but when I try…
ExodusSolis
  • 23
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Help Evaluating Legendre Integral times Polynomial

In calculating the multipole moments on the Maclaurin spheroid, I've encountered the following integral. The text I'm reading (Poisson and Will) says that the integral is doable. $$ \int_{-1}^{1} \left(1+a x^2\right)^{-\frac{l+3}{2}} P_l(x) d…
malxmusician212
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How to calculate this definite integral

Can someone explain how is the result of: $A(x_0,y_0,z_0)=\int\limits_{-a}^{a}\int\limits_{-b}^{b}\int\limits_{-\infty}^{+\infty}\frac{1}{\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}}{\rm d}z{\rm d}y{\rm d}x$ equal to (assuming…
user89723
  • 11
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Integral in two variables of intersection of two circumference

I have to resolve the integral $$\iint_{\Omega} x \sqrt{x^2+y^2} dxdy$$ with $\Omega=\{(x,y)\in \mathbb{R}^2| x^2+y^2<1, x^2+y^2<2y, x<0 \}$ The geometric interpretation of this set is not to difficult, the first information is a ball of radius $1$…
Mario
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An integral involving a logarithm and two inverse trigonometric functions:$\displaystyle \int_0^1 \arcsin x \arctan x\log x dx$

The integral $\displaystyle\int_0^1 \arcsin x \arctan x\log x dx$ can be explicitly computed; the result being representable in an elementary way (for instance without the appearance of polylogarithms). This was done around 2017. I would be…
ray
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Integrating $\int_0^1 \frac{1}{\sqrt{1-y^4}}dy$

What do I need to manipulate to show that $$\int_0^1 \frac{1}{\sqrt{1-y^4}}dy=\frac 1 4 \int_0^1 t^{-3/4}(1-t)^{-1/2}dt$$
hasExams
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Help for the integral of trigonometric function Gaussian kernel

Can anyone give me some clues about the following integral \begin{equation*} \int_{R^{n}} \cos(\|x\|)\exp(-0.5(x - Ex)'C^{-1}(x - Ex))dx \end{equation*} where $\|x\|$ is the Euclidean norm. I only know the following special case \begin{equation*} …
hmeng
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$\int_{0}^{1} \left(1-x^a\right)\cdot \log\left(1-x^a \right)\,dx$ for a positive

I want to calculate $$\int_{0}^{1} \left(1-x^a\right)\cdot \log\left(1-x^a \right)\,dx,$$ but no luck. Wolfram gives me a very complicated answer in terms of hypergeometric functions. Is there a better answer or a way to tackle this to get a…
Jama
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Integrating a rational function with a geometric series

$$\int_{0}^{1}\dfrac{x^{46}}{\sum_{n=0}^{100}x^n}\mathrm{d}x$$ How to integrate the above expression? I am trying the above question by breaking the denominator. I broke the denominator like $\dfrac{1-x^{101}}{1-x}$. Now $(1-x)$ will go into the…
DEEPANWITA ROY
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Why does $\int_{0}^{1} \mathbb{1}^x_{(-y,y)}dy = \int_{0}^{1} \mathbb{1}^y_{(|x|,1)}dy $?

Let $\mathbb{1}$ be the indicator function, e.g $$\mathbb{1}^x_{(0,1)} = \begin{cases} 1, \ x \in (0,1) \\ 0, \ \text{elsewhere}. \end{cases}$$ I've stumbled onto this integral, $\int_{0}^{1} \mathbb{1}^x_{(-y,y)}dy$. For some (obvious) reason, this…
Oskar
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Finding Total number of positive continuous function in $[0,1]$

Finding Total number of positive continuous function $g(x)$ in $[0,1]$ which satisfy $\displaystyle \int^1_{0}g(x)dx=1,\;\int^{1}_{0}xg(x)dx=2\,\ \int^1_0x^2g(x)dx=4.$ What I try : I am Trying to solve using Cauchy schwarz Inequality Using…
jacky
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Integral of $|\sin x|$ from $0$ to $\pi$

Logically because $\sin$ is positive in the given limits, I thought the answer would be the same as integral of $\sin x$ for the given limits (without mod). By calculating it that way I got $2$ which is wrong according to the answers. What did I do…
Silverstreak
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How to evaluate the following integral?

How do I evaluate the following expression? $$\int_1^3\frac{1+x}{1+x^3}~\mathrm{d}x$$ I am trying this question by integrating the expressions separately like $\frac{1}{1+x^3}$ and $\frac{x}{1+x^3}$. But can't solve after that. How to proceed…
SUBHA SANKAR ROY
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An integral of $\displaystyle\int_{0}^{1}{\ln\left(1+\dfrac{\ln^2x}{4π^2}\right)\dfrac{\ln(1-x)}{x}\ \mathrm{d}x}$

I have two forms of solution for this integral: First : $$\displaystyle \begin{aligned}\int_{0}^{1}{\ln\left(1+\dfrac{\ln^2x}{4π^2}\right)\dfrac{\ln(1-x)}{x}\ \mathrm{d}x}&=2π\int_{0}^{\infty}{\ln(1+x^2)\ln(1-e^{-2πx})\…
Dylan Lee
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