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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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Definite integral $\int_{-2}^{2} \frac{x^2}{1+5^x}\,\mathrm{d}x$

The question is to find the value of the definite integral: $$I=\int_{-2}^{2} \frac{x^2}{1+5^x}\,\mathrm{d}x.$$ This question appeared in this year's CBSE board exam. Attempts: Replace $x\to-x$ and we get $$I=\int_{-2}^{2} 5^x…
Kartik
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Checking how to solve $\int_{-\infty}^\infty x^2/e^{ax^2} dx$

I used Gamma function in the form Gamma$(z)=2\int_0^\infty e^{-t^2}t^{2z-1}dt$ which yields $ \pi^{1/2}/2$ for $z=3/2$. Thus, $\int_{-\infty}^\infty x^2/e^{ax^2} dx =2\int_0^\infty x^2/e^{ax^2} dx = \pi^{1/2}/2a^{3/2}$ right? Did I misuse the Gamma…
Patrick
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Revolution of a solid volume around the $y$-axis

I need to calculate the volume between $y = x$ and $x = 4y - y^2$ around the $y$-axis I am not sure how I can not convert the $x = 4y - y^2$ to being of the form $y = \cdots$ so that I could use it in the shell method: $V = \int 2\pi\times…
Nicolas
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Definite Integral..Clarify my work..

My professor worked this solution in class, but I am not sure he is right. I did it on my own if someone could take a look at my work. I am a little confused. $$\int_1^2 \frac{\ln(x)}{x} \;dx$$ So I approached this problem with $U$ Substitution.…
Joe Caraccio
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Prove that $\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$.

Prove that $$\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$$ for all $k\in\mathbb N$ and $h$ even integer. I have tried with Induction Principle (for $h$) but without success.
Mark
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Is there an error in my textbook?

The weird thing is, the last sentence says, for the case where $x_1$, "since $\Delta x_1$ approaches zero as $\max \Delta x_k \rightarrow 0...$". But given that $\Delta x_1$ is formed between $x_0$ and $x_1$, and if $x_1$ is equal to $0$ and the…
most venerable sir
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Prove that the following statement is true

$$ \int_{0}^1 e^{x^2} dx+\int_{1}^e \sqrt {lnx} dx=e$$ Prove the above mentioned without using the imaginary function error "erfi".Solve it by using basic definite integrals properties.
Razvan Paraschiv
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Check this double integral

I'm learning solid angles from this page among others. Half way down it shows a double integral: $$\int_{\phi_1}^{\phi_2} d \phi\int_{\theta_1}^{\theta_2}\sin \theta\,d\theta = (\phi_2 - \phi_1)(\cos \theta_2 - \cos \theta_1)$$ I think the…
PeteUK
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Finding area of enclosed region of y = mx and y = $\frac {x}{x^2+10} $

Homework Problem Statement: Any line of the form $y = mx$ will intersect the curve $y = \frac {x}{x^2+10} $ in precisely three points provided $0 < m < B$ for some number $B$. What is the value of $B$? When the line $y = mx$ intersects the curve in…
LRo
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Area bounded by $(x^2+y^2)^2 = a(x^3-3xy^2)$

I'm suppose to calculate the area bounded by the curve $(x^2+y^2)^2 = a(x^3-3xy^2)$ and my guess was to convert this equation into polar coordinate (x=rcos$\theta$, y=rsin$\theta$ and $r^2=x^2+y^2$). On doing so I obtained the equation, $r^4 = a…
rndflas
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An exponential integral

How to simplify the integral given below? $$ \int_{0}^{\infty}\frac{xe^{-\beta x}}{k^{x+1}}\mathrm{d}x,\quad k=1+e^{-a},\ a\in\mathbb{R},\;\beta>0 $$ Is there an explicit form, or can it be expressed with summation symbol? Do i need to use gamma…
mert
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Definite Integral Evaluation Undefined

We have $$ f(x) := \frac{e^{1/x}}{x^2}, \qquad x \ne 0$$ We need to determine a number $a<0$ such that $$ \int^0_a f(x)\, dx = f(a). $$ What I tried: With the substitution technique I get to $[-e^{1/x}]^0_a$. So that means we would get $-e^{1/0}…
John Snoe
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Integral $\int_{-\pi}^\pi e^{\sin x} \sin 4x \ dx$

I found this question in an old real analysis text book, (so old the cover had come off) I graphed it on wolfram alpha, and it looks like an almost odd function. I think the integral evaluates to zero. Would anyone care help prove? (assuming I'm…
ptr64
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How apply Leibinz Rule in $\frac{\partial^2}{\partial t^2}\Big\{ \int_0^t \int_{x-c(t-\tau))}^{x+c(t-\tau)} f(s, \tau)ds \, d\tau \Big\}$

$$\frac{\partial^2}{\partial t^2}\Big\{ \int_0^t \int_{x-c(t-\tau))}^{x+c(t-\tau)} f(s, \tau)ds \, d\tau \Big\}$$ I have seen examples as a start point, but i can't find one like it. Any tip will be apreciated
Juan Camilo Acosta Arango
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Calculate $\int_{-\pi}^\pi\delta(\omega-\omega_0) e^{j\omega t}d\omega$.

Let us consider the Dirac delta function $\delta(\omega)$ Calculate: $$\int_{-\pi}^\pi\delta(\omega-\omega_0) e^{j\omega t}d\omega$$ The presence of Delta function gives me some problems. I would appreciate some help with this problem.
Lely
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