CPM Homework Help : CALC Problem 6-61

Examine the integrals below. Consider the multiple tools available for evaluating integrals and use the best strategy for each. After evaluating the integral, write a short description of your method.

$\int _ { \pi / 6 } ^ { \pi / 4 } ( \operatorname { csc } ^ { 2 }( x )) d x$
Hint (a):
Think! What function has a derivative of $y=\operatorname{csc}^2x$ ? Be mindful about positive and negative signs.

$\int ( 5 ^ { y } - 3 ( 2 y + 1 ) ^ { 2 } \cdot 2 ) d y$
Hint (b):
Before you integrate, expand the integrand.

$\int _ { 0 } ^ { \pi / 4 } \operatorname { tan }( t )d t$
Hint (c):
Use your calculator. (Unless you know what the antiderivative of $y =\operatorname{tan}x$ is).

$\int ( \operatorname { tan } ^ { 2 } (y) + 1 ) d y$
Steps (d):
Continue to simplify the integrand, and then evaluate.