### Home > APCALC > Chapter 8 > Lesson 8.2.1 > Problem8-69

8-69.

Examine the integrals below. Consider the multiple tools available for evaluating integrals and use the best strategy for each. Evaluate each integral and briefly describe your method.

1. $\int _ { 1 } ^ { 4 } ( 2 \sqrt { t } + t ^ { 2 } ) d t$

Before integrating, rewrite the integrand using a fractional exponent.

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

Let $U = x^2 + 1$.

Because $y = x^2 + 1$ is always positive, this solution will not need an absolute value sign.

1. $\int _ { - \pi / 4 } ^ { \pi / 4 } \operatorname { tan } ( u ) d u$

Think about the shape of this graph, does it have any symmetry?

1. $\int \operatorname { ln } ( e ^ { 2 x } ) d x$

Before integrating, use properties of logarithms to rewrite the integrand.

$=\int2xdx=x^2+C$