The positive integer n leq 5 for which int_0^ | vedclass

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

Question

(C) Let $I_n = \int_0^1 e^x(x-1)^n dx$. Using integration by parts,we have $\int u dv = uv - \int v du$. Let $u = (x-1)^n$ and $dv = e^x dx$. Then $du

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(C) Let $I_n = \int_0^1 e^x(x-1)^n dx$. Using integration by parts,we have $\int u dv = uv - \int v du$. Let $u = (x-1)^n$ and $dv = e^x dx$. Then $du = n(x-1)^{n-1} dx$ and $v = e^x$. $I_n = [e^x(x-1)^n]_0^1 - n \int_0^1 e^x(x-1)^{n-1} dx = (0 - (-1)^n) - n I_{n-1} = (-1)^{n+1} - n I_{n-1}$. For $n=1$: $I_1 = (-1)^2 - 1 I_0 = 1 - \int_0^1 e^x dx = 1 - (e-1) = 2-e$. For $n=2$: $I_2 = (-1)^3 - 2 I_1 = -1 - 2(2-e) = -1 - 4 + 2e = 2e-5$. For $n=3$: $I_3 = (-1)^4 - 3 I_2 = 1 - 3(2e-5) = 1 - 6e + 15 = 16-6e$. Comparing this with the given value $16-6e$,we find $n=3$.

The positive integer $n \leq 5$ for which $\int_0^1 e^x(x-1)^n dx = 16-6e$ is

Similar Questions

$\int \sec ^{-1} x \, dx =$

$\int \sin^{-1} x \, dx$ is equal to

$\int x^3(\log x)^2 \, dx = $

$\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x=$

$\int (x^2 + 3x + 2) e^x dx = $ . . . . . . $+ C$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module