Let be any differentiable function of .

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

g(x)

be any differentiable function of

x

. (i) Prove that

\dfrac{d}{dx}\bigl[\cos g(x)\bigr]= - g'(x)\,\sin g(x).

(ii) Using the result of part (i), evaluate

\displaystyle \int 4x^{3}\,\sin\bigl(x^{4}+5\bigr)\,dx.

(iii) Differentiate your answer to part (ii) to verify that it is indeed an antiderivative of the given integrand.