The value of int_0^1 {x^2 e^x dx} is equal to | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $I = \int_0^1 x^2 e^x dx$. Using integration by parts,$\int u v dx = u \int v dx - \int (u' \int v dx) dx$. Let $u = x^2$ and $v = e^x$. Then

Vedclass · Accepted Answer

$e - 2$

Vedclass · Answer

(A) Let $I = \int_0^1 x^2 e^x dx$. Using integration by parts,$\int u v dx = u \int v dx - \int (u' \int v dx) dx$. Let $u = x^2$ and $v = e^x$. Then $u' = 2x$ and $\int v dx = e^x$. $I = [x^2 e^x]_0^1 - \int_0^1 2x e^x dx$. $I = (1^2 e^1 - 0^2 e^0) - 2 \int_0^1 x e^x dx$. $I = e - 2 \int_0^1 x e^x dx$. Now,evaluate $\int_0^1 x e^x dx$ using integration by parts again: $\int_0^1 x e^x dx = [x e^x]_0^1 - \int_0^1 e^x dx = (e - 0) - [e^x]_0^1 = e - (e - 1) = 1$. Substituting this back into the expression for $I$: $I = e - 2(1) = e - 2$.

The value of $\int_0^1 {x^2 e^x dx}$ is equal to

Similar Questions

$\int_0^\pi \sqrt{1+4 \sin^2 \frac{x}{2}+4 \sin \frac{x}{2}} \, dx$ is equal to

$\int_{0}^{2} |x - 1| \, dx = $

$\int_{0}^{\frac{\pi}{2}} \sqrt{\cos \theta} \cdot \sin^{3} \theta d \theta = . . . . . .$

$\int_{0}^{1} \frac{d}{dx} \left[ \sin^{-1} \left( \frac{2x}{1+x^2} \right) \right] dx$ is equal to

$ \int_{0}^{1} \sqrt{\frac{1+x}{1-x}} \, dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module