91Ó°ÊÓ

(a) Make the indicated \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{\sin ^{2}(\ln x)}{x} d x, u=\ln x$$

Evaluate the integral. $$ \int \frac{x^{3}-2 x^{2}+2 x-2}{x^{2}+1} d x $$

Determine whether the statement is true or false. Explain your answer. $$ \int_{1}^{+\infty} x^{-4 / 3} d x \text { converges to } 3 $$

(a) Derive the identity $$ \frac{\sec ^{2} x}{\tan x}=\frac{1}{\sin x \cos x} $$ (b) Use the identity \(\sin 2 x=2 \sin x \cos x\) along with the result in part (a) to evaluate \(\int \csc x d x\) (c) Use the identity \(\cos x=\sin [(\pi / 2)-x]\) along with your answer to part (a) to evaluate \(\int \sec x d x\)

Evaluate the integral. $$\int \sec ^{5} x \tan ^{3} x d x$$

Evaluate the integral. $$ \int_{2}^{4} \sec ^{-1} \sqrt{\theta} d \theta $$

Evaluate the integral. $$ \int_{1}^{2} x \sec ^{-1} x d x $$

Find the arc length of the curve \(y=x^{2}\) from \(x=0\) to \(x=1\)

In each part, determine whether a trapezoidal approximation would be an underestimate or an overestimate for the definite integral. $$ \begin{array}{lll}{\text { (a) } \int_{0}^{1} \cos \left(x^{2}\right) d x} & {\text { (b) } \int_{3 / 2}^{2} \cos \left(x^{2}\right) d x} & {}\end{array} $$

Evaluate the integral. $$ \int \frac{x^{4}+6 x^{3}+10 x^{2}+x}{x^{2}+6 x+10} d x $$