91Ó°ÊÓ

Find the integrals $$\int \cos ^{2}(3 \alpha+1) d \alpha$$

Use Table 7.9 $$\begin{array}{c|c|c|c|c|c} \hline t & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 \\ \hline g(t) & 1.87 & 2.64 & 3.34 & 3.98 & 4.55 \\ \hline t & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\ \hline g(t) & 5.07 & 5.54 & 5.96 & 6.35 & 6.69 \\\ \hline \end{array}$$ Estimate \(\int_{0.2}^{0.6} g(t) d t\) using a left-hand sum with \(n=4.\)

Decide if the improper integral converges or diverges. $$\int_{50}^{\infty} \frac{d z}{z^{3}}$$

Say which formula, if any, to apply from the table of integrals. Give the values of any constants. $$\int \frac{4 x-2}{x^{2}-9} d x$$

Find the integrals Check your answers by differentiation. $$\int t^{2}\left(t^{3}-3\right)^{10} d t$$

Calculate the integral if it converges. You may calculate the limit by appealing to the dominance of one function over another, or by l'Hopital's rule. $$\int_{-\infty}^{\infty} \frac{d z}{z^{2}+25}$$

Calculate the integral if it converges. You may calculate the limit by appealing to the dominance of one function over another, or by l'Hopital's rule. $$\int_{1}^{\infty} \frac{z}{\left(1+z^{2}\right)^{3}} d z$$

Decide if the improper integral converges or diverges. $$\int_{1}^{\infty} \frac{d x}{1+x}$$

Find the integrals Check your answers by differentiation. $$\int x\left(x^{2}+3\right)^{2} d x$$

Evaluate the integral.$$\int \frac{3 x^{2}-8 x+1}{x^{3}-4 x^{2}+x+6} d x ; \text { use } \frac{A}{x-2}+\frac{B}{x+1}+\frac{C}{x-3}$$.