Answer to Question #327471 in Calculus for Nur

Question

1.Find ∫cos^2 (2x)sin x dx

2.Find ∫(b − ax)e^4x dx . Give your answer in factor form.

Answer 1

1.$\int cos^2(2x)sin(x)dx=-\int cos^2(2x)d(cos(x))$. Make a change of variable $z=cos(x)$. We receive:

$-\int(2z^2-1)dz=-(\frac23z^3-z)+C=-\frac23(cos(x))^3+cos(x)+C,C\in{\mathbb{R}}$

$2.\,\int(b-ax)e^{4x}dx=\frac14be^{4x}-a\int xe^{4x}dx=\frac{1}{4}be^{4x}-a(\frac14xe^{4x}-\int \frac14e^{4x}dx)+C=\frac{1}{4}be^{4x}-a(\frac14xe^{4x}-\frac{1}{16}e^{4x}dx)+C,C\in{\mathbb{R}}$