  ### Home > APCALC > Chapter 8 > Lesson 8.1.2 > Problem8-18

8-18.

No calculator! Evaluate each of the following integrals. 1. $\int \frac { 8 x ^ { 3 } - 1 } { 6 x ^ { 4 } - 3 x } d x$

$u$-substitution

Let $u = 6x^{4} − 3x$.

1. $\int _ { 2 } ^ { 5 } \pi ( ( x + 1 ) ^ { 2 } - 3 ^ { 2 } ) d x$

Before you integrate, factor the $π$ out of the integrand. You may (or may not) choose to expand $(x + 1)^2$.

1. $\int ( 5 x - 2 + \frac { 5 } { x + 3 } ) d x$

$\text{Recall that }\frac{5}{x+3}\text{ is a transformation of }\frac{1}{x},$

$\text{ and you know the antiderivative of }\frac{1}{x}.$

1. $\int 2 x \operatorname { sin } ( 11 x ^ { 2 } - 3 ) d x$

$u$-substitution

1. $\int _ { - 1 } ^ { 0 } \frac { 2 x } { x + 2 } d x$

You could long divide first or use $u$-substitution.
Let $u =$ the denominator.

If $u = x + 2$ then $x = u - 2$.

$\text{Therefore numerator }2x=2(u-2) \text{ and }\frac{du}{dx}=1 \text{ so }du=dx.$

Also bounds $u(0) = 2$ and $u(-1) = 1$.
Rewrite the integral and evaluate.

1. $\int \frac { 3 } { \sqrt { 1 - 9 x ^ { 2 } } } d x$

Recall that $9x^2 = (3x)^2$.
Then look for a familiar antiderivative.