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

8-18.

No calculator! Evaluate the following integrals.

1. $\int \frac { 8 x ^ { 3 } - 1 } { 6 x ^ { 4 } - 3 x } d x$

$U$-substitution.

$U = 6x^4 − 3x$

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

Recall that $\frac{5}{x+3}$ is a transformation of $\frac{1}{x}$, 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 $x = U-2$.
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. 