Let be a real constant and define the function i(k)= _ 0 ^ 4 e^ k x ,dx It is known that .

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

k

be a real constant and define the function i(k)=\int_{0}^{\ln 4} e^{k x}\,dx It is known that

I(k)=10

. (a) Evaluate

I(k)

in terms of

k

(simplify completely). (b) Using the given value

I(k)=10

, determine all real values of

k

that satisfy the equation. (c) Verify your solution(s) for

k

by direct substitution into the original integral. (d) Explain what the sign of the constant

k

implies about the behavior of the integrand

e^{k x}

on the interval

[0,\ln 4]