Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Question

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.

Answer 1

You may just perform the change of variable $u=2+x^{9/2}$, $du=\dfrac92x^{7/2}\:dx$, giving $$ \int x^{7/2} \sec^2(2+x^{9/2})\: \mathrm{d}x=\dfrac29\int \sec^2(u) \:\mathrm{d}u= \dfrac29\tan (u)+C. $$ Can you take it from here?

Answer 2

Substitute $u=$ the term inside the parentheses.