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Use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{3}-8 x^{2}-1}{(x+3)\left(x^{2}-4 x+5\right)} d x $$

Use integration by parts to evaluate each integral. \(\int x(3 x+10)^{49} d x\)

Evaluate each integral. \(\int e^{\ln (3 \cos x)} d x\)

Use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{\left(\sin ^{3} t-8 \sin ^{2} t-1\right) \cos t}{(\sin t+3)\left(\sin ^{2} t-4 \sin t+5\right)} d t $$

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, give a numerical approximation. $$\int_{0}^{\pi / 2} \sin ^{12} x d x$$

Perform the indicated integrations. \(\int \frac{t^{2} \cos \left(t^{3}-2\right)}{\sin ^{2}\left(t^{3}-2\right)} d t\)

Show that $$ \lim _{n \rightarrow \infty} \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cdots \cos \frac{x}{2^{n}}=\frac{\sin x}{x} $$ by completing the following steps. $$ \begin{array}{l} \text { (a) } \cos \frac{x}{2} \cos \frac{x}{4} \cdots \cos \frac{x}{2^{n}}= \\\ {\left[\cos \frac{1}{2^{n}} x+\cos \frac{3}{2^{n}} x+\cdots+\cos \frac{2^{n}-1}{2^{n}} x\right] \frac{1}{2^{n-1}}} \end{array} $$ (See Problem 46 of Section \(0.7 .)\) (b) Recognize a Riemann sum leading to a definite integral. (c) Evaluate the definite integral.

Use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{\cos t}{\sin ^{4} t-16} d t $$

Evaluate each integral. \(\int \frac{\sqrt{9-y^{2}}}{y} d y\)

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, give a numerical approximation. $$\int_{0}^{\pi} \cos ^{4} \frac{x}{2} d x$$