91Ó°ÊÓ

Evaluate the integral. $$ \int \frac{d x}{x^{2}-4 x+5} $$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$ \int_{0}^{+\infty} \frac{e^{-\sqrt{x}}}{\sqrt{x}} d x ; u=\sqrt{x} \quad[\text { Note }: u \rightarrow+\infty \text { as } x \rightarrow+\infty .] $$

(a) Make an appropriate \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a). $$ \int \frac{\cos 3 x}{(\sin 3 x)(\sin 3 x+1)^{2}} d x $$

Evaluate the integral. $$ \int_{1}^{3} \sqrt{x} \tan ^{-1} \sqrt{x} d x $$

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

True-False Determine whether the statement is true or false. Explain your answer. The partial fraction decomposition of $$ \frac{2 x+3}{x^{2}} \text { is } \frac{2}{x}+\frac{3}{x^{2}} $$

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

True-False Determine whether the statement is true or false. Explain your answer. If \(f(x)=P(x) /(x+5)^{3}\) is a proper rational function, then the partial fraction decomposition of \(f(x)\) has terms with constant numerators and denominators \((x+5),(x+5)^{2}\) and \((x+5)^{3}\).

Evaluate the integral. $$ \int_{0}^{2} \ln \left(x^{2}+1\right) d x $$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$ \int_{12}^{+\infty} \frac{d x}{\sqrt{x}(x+4)} ; u=\sqrt{x} \quad[\text { Note }: u \rightarrow+\infty \text { as } x \rightarrow+\infty .] $$