Let with constant and integer

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

Let

K_n=\displaystyle\int x^{n}e^{kx}\,dx

with constant

k\ne 0

and integer

n\ge 1

. Which of the following identities correctly relates

K_n

K_{n-1}

? (a)

K_n=\dfrac{x^{n}e^{kx}}{k}+\dfrac{n}{k}\,K_{n-1}

(b)

K_n=\dfrac{x^{n}e^{kx}}{k}-\dfrac{n}{k}\,K_{n-1}

(c)

K_n=\dfrac{x^{n}e^{kx}}{n}-\dfrac{k}{n}\,K_{n-1}

(d)

K_n=\dfrac{x^{n}e^{kx}}{k}-\,K_{n-1}