Let be a continuous function and let denote an antiderivative of such that for all…

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

Problem

Let

f

be a continuous function and let

F

denote an antiderivative of

f

such that

F'(u)=f(u)

for all real numbers

u

. Suppose

g

is a differentiable function on

[a,b]

. (a) Prove that

\dfrac{d}{dx}\bigl[F(g(x))\bigr]=f(g(x))\,g'(x)

for all

x

[a,b]

. Clearly state any theorem you use. (b) Use the result from part (a) to justify the relationship

\int_{a}^{b}f(g(x))\,g'(x)\,dx=F\bigl(g(b)\bigr)-F\bigl(g(a)\bigr).

Describe how the limits of integration change when the variable of integration is switched from

x

u=g(x)

.},{ (c) Evaluate

\int_{0}^{1}\cos\bigl(x^{3}\bigr)\,3x^{2}\,dx

using the relationship established in part (b).