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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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Solution to the definite integral $\int_0^{\infty}x^{-\alpha} e^{-\beta/x}\,dx$

Is there a explicit solution to this definite integral, with $\alpha>0$ and $\beta>0$ : $$\int_0^{\infty}x^{-\alpha} e^{-\beta/x}\,dx$$ If yes, what is it?
Jose Soto
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Definite integral $\int ^b _a f(x)dx = \lim_{n \to \infty} \sum^n _{k=1} f(\zeta_k) \Delta x_k$

This is the mathematical description of the definite integral of $f(x)$ between $a$ and $b$: $$\int ^b _a f(x)dx = \lim_{n \to \infty} \sum^n _{k=1} f(\zeta_k) \Delta x_k$$ In here, $\Delta x_k$ is the width of a rectangle, $f(\zeta_k)$ is the…
Integrals
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Evaluation of inverse of $tan$ integration

Evaluation of $$\int^{\frac{\pi}{2}}_{0}\frac{x\cos x}{1+\sin^2 x}dx$$ Try: Integration By parts $$I = x\tan^{-1}(\sin x)\bigg|^{\frac{\pi}{2}}_{0}-\int^{\frac{\pi}{2}}_{0}\tan^{-1}(\sin x)dx$$ $$I =…
DXT
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How to demonstrate $ \int\limits_1^{N+1}f(x)dx\leq \sum\limits_{k=1}^{N}f(k)\leq \int\limits_0^N f(x)dx $ for $f$ decreasing?

I found this inequality in an exercises here, on this site (see the inequality below). It applies to a decreasing function. I used this for my exercise and it worked. The problem is i cannot find from where this comes. Can someone explain it to me ?…
Alex Hrenciuc
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Integration of looped sin function

At infinity, i took limits to be L. so L= sin (x+L) then after integration i am left withL= 2 cosL. how to get value of L
maveric
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Evaluating $\int_{-5 }^{-2} f(x)\;dx + \int_{1/6}^{1/3}f(x)\,dx+\int_{6/5}^{3/2}f(x)\;dx$, where $f(x)=\left(\frac{x^2-x}{x^3-3x+1}\right)^2$

How to find the value of $$\int _ { - 5 } ^ { - 2 } \left( \frac { x ^ { 2 } - x } { x ^ { 3 } - 3 x + 1 } \right) ^ { 2 }\,dx + \int _ { \frac 16 } ^ { \frac 13 } \left( \frac { x ^ { 2 } - x } { x ^ { 3 } - 3 x + 1 } \right) ^ { 2 }\,dx+\int _ {…
maveric
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Definite integration when denominator consists of $x\sin x +1$

$$\int_0^\pi\frac{x^2\cos^2x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx$$ The answer is $0$. I tried and made $(x\sin x +1)^2 $ in numerator and proceed, but not able to do any further.
maveric
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Typo in my textbook about integrals?

I am reading about integrals: The right hand side of the last equation reads: $$ ... x^3 \cdot 1 \cdot x \cdot \sin x... $$ Is that $1$ supposed to be a $+$ instead?
Jwan622
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Finding function $f(x),$ if some conditions are given

If $f(x)$ is a defined in $[0,1]$ such that $\displaystyle \int^{1}_{0}(f(x))^2\,dx=4$ and $\displaystyle \int^{1}_{0}f(x)\,dx=\int^{1}_{0}x\cdot f(x)\,dx=1,$ then what is the value of $\displaystyle \int^{1}_{0}(f(x))^3\,dx?$ Try: First thing in…
DXT
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Conversion a three-dimensional integral into a one-dimensional one

I want to convert this integral $$\int_{\mathbb{R}^{3}}d^{3}k\frac{1}{|\mathbf{k}|^{2}}e^{i\mathbf{k}\cdot(\mathbf{x}-\mathbf{x}')}$$ to a one-dimensional one (here $d^{3}k$ means an element of volume of $\mathbb{R}^{3}$). I found a formula, but I…
Ana S. H.
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Equation with definite integral.

Suppose that I have a function $f(x,y)$, I want to solve $$\int_a^bf(x,y)dx =0.$$ w.r.t $y$. Is it possible to say that this is equivalent to solve $$f(b,y)-f(a,y) =0$$ since I can multiply on both the sides by $\frac{d}{dx}$?
Sam
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How does one prove the midpoint rule for triple integral?

How does one prove the midpoint rule for triple integral? That is, $$\sum_i \sum_j \sum_k f(\hat{x_i}, \hat{y_j}, \hat{z_k}) \Delta x_i \Delta y_j \Delta z_k$$ where $\displaystyle (\hat{x_i}, \hat{y_i}, \hat{z_i})=\bigg( \frac{x_{i-1}+x_i}{2},…
mavavilj
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Integral using polar coordinates

Let $X=\{(x,y)\in\mathbf{R}^2\mid x^2+y^2\leqslant 1,x,y\geqslant 0 \}$. Calculate $\int_X xye^{x^2+y^2}\,dx\,dy$. By using polar coordinates, I get $=\int_0^{\pi/2}\int_0^1e \cos\theta\sin\theta\,dr\,d\theta=\int_0^{\pi/2}\frac{1}{2}e\sin…
Dr. Heinz Doofenshmirtz
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Length of a plane curve

Find the length of the curve $$x = t - sin\,t, \quad\quad y = 1 - cos\,t, \quad\quad 0\leq t\leq 2\pi$$ I have gotten close to the solution, but clearly I am making an error because I arrive at 0 for the answer. $$\frac{dx}{dt}=1-cos\,t, \quad…
JMSE
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Length of $r=1+\sin{\theta}$ and the interval

Find the length of $r=1+\sin{\theta}$. I got to $\sqrt{2} \int\limits_{0}^{2\pi}\sqrt{1+\sin{\theta}} \,\mathrm{d}\theta$. And the first way I used to solve the integral was substitution of $1+\sin{\theta}=u$. Thus $ \cos\theta…
nik
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