91Ó°ÊÓ

\(1-80\) Evaluate the integral. $$\int \frac{\sin ^{3} x}{\cos x} d x$$

\(1-80\) Evaluate the integral. $$\int \frac{\sin x+\sec x}{\tan x} d x$$

\(3-32\) Evaluate the integral. $$\int x \cos 5 x d x$$

Evaluate the integral. $$\int_{\pi / 2}^{3 \pi / 4} \sin ^{5} x \cos ^{3} x d x$$

I-6 Write out the form of the partial fraction decomposition of the function (as in Example \(7 ) .\) Do not determine the numerical values of the coefficients. (a) \(\frac{x^{4}+1}{x^{5}+4 x^{3}} \quad\) (b) \(\frac{1}{\left(x^{2}-9\right)^{2}}\)

Estimate \(\int_{0}^{1} \cos \left(x^{2}\right) d x\) using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with \(n=4 .\) From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you conclude about the true value of the integral?

Find the area under the curve \(y=1 / x^{3}\) from \(x=1\) to \(x=t\) and evaluate it for \(t=10,100,\) and \(1000 .\) Then find the total area under this curve for \(x \geq 1\)

\(1-3\) Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. $$\int \frac{X^{3}}{\sqrt{x^{2}+9}} d x ; \quad x=3 \tan \theta$$

Evaluate the integral. $$\int_{0}^{\pi / 2} \cos ^{5} x d x$$

(a) Graph the functions \(f(x)=1 / x^{1.1}\) and \(g(x)=1 / x^{0.9}\) in the viewing rectangles \([0,10]\) by \([0,1]\) and \([0,100]\) by \([0,1]\). (b) Find the areas under the graphs of \(f\) and \(g\) from \(x=1\) to \(x=t\) and evaluate for \(t=10,100,10^{4}, 10^{6}, 10^{10}\), and \(10^{20}\). (c) Find the total area under each curve for \(x \geqslant 1\), if it exists.