### Home > APCALC > Chapter 6 > Lesson 6.2.1 > Problem6-61

6-61.

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

1. $\int _ { \pi / 6 } ^ { \pi / 4 } ( \operatorname { csc } ^ { 2 } ( x ) ) d x$

Think! What function has a derivative of $y = \csc^²x$? Be mindful about positive and negative signs.

1. $\int ( 5 ^ { y } - 3 ( 2 y + 1 ) ^ { 2 } \cdot 2 ) d y$

Before you integrate, expand the integrand.

1. $\int _ { 0 } ^ { \pi / 4 } \operatorname { tan } ( t ) d t$

Use your calculator. (Unless you know what the antiderivative of $y = \tan x$ is).

1. $\int ( \operatorname { tan } ^ { 2 } ( y ) + 1 ) d y$

$\int (\text{tan}^{2}y+1)dy=\int \frac{\text{sin}^{2}y}{\text{cos}^{2}y}+\frac{\text{cos}^{2}y}{\text{cos}^{2}y}dy=$

Continue to simplify the integrand, and then evaluate.