91Ó°ÊÓ

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \csc ^{2} 2 \theta \cot 2 \theta d \theta $$ $$ \text {a.}u=\cot 2 \theta \quad \text { b. Using } u=\csc 2 \theta $$

Evaluate the integrals $$ \int_{0}^{\pi / 4} \tan ^{2} x d x $$

In Exercises \(15-18,\) use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$f(x)=x^{3} \text { on } [0,2]$$

Evaluate the integrals $$ \int_{0}^{\pi / 6}(\sec x+\tan x)^{2} d x $$

Express the sums in sigma notation. The form of your answer will depend on your choice for the starting index. $$ -\frac{1}{5}+\frac{2}{5}-\frac{3}{5}+\frac{4}{5}-\frac{5}{5} $$

In Exercises \(15-18,\) use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$f(x)=1 / x \text { on } [1,9]$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \frac{d x}{\sqrt{5 x+8}} $$ $$ \text {a.} u=5 x+8 \quad \text { b. Using } u=\sqrt{5 x+8} $$

In Exercises \(15-22,\) graph the integrands and use known area formulas to evaluate the integrals. $$ \int_{1 / 2}^{3 / 2}(-2 x+4) d x $$

In Exercises \(15-18,\) use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$f(t)=(1 / 2)+\sin ^{2} \pi t \text { on } [0,2]$$

Evaluate the integrals $$ \int_{0}^{\pi / 8} \sin 2 x d x $$