91Ó°ÊÓ

Show that if \(f\) is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value of \(\int_{a}^{b} f(x) d x\)

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

Use a CAS to find an antiderivative \(F\) of \(f\) such that \(F(0)=0 .\) Graph \(f\) and \(F\) and locate approximately the \(x\) -coordinates of the extreme points and inflection points of \(F .\) $$f(x)=x e^{-x} \sin x, \quad-5 \leqslant x \leqslant 5$$

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

\(1-80\) Evaluate the integral. $$\int x^{3}(x-1)^{-4} d x$$

\(\begin{array}{l}{39-50 \text { Make a substitution to express the integrand as a rational }} \\ {\text { function and then evaluate the integral. }}\end{array}\) $$ \int \frac{e^{2 x}}{e^{2 x}+3 e^{x}+2} d x $$

Use a CAS to find an antiderivative \(F\) of \(f\) such that \(F(0)=0 .\) Graph \(f\) and \(F\) and locate approximately the \(x\) -coordinates of the extreme points and inflection points of \(F .\) $$f(x)=\sin ^{4} x \cos ^{6} x, \quad 0 \leqslant x \leqslant \pi$$

(a) If \(g(x)=\left(\sin ^{2} x\right) / x^{2},\) use your calculator or computer to make a table of approximate values of \(\int_{1}^{t} g(x) d x\) for \(t=2,5,10,100,1000,\) and \(10,000 .\) Does it appear that \(\int_{1}^{\infty} g(x) d x\) is convergent? (b) Use the Comparison Theorem with \(f(x)=1 / x^{2}\) to show that \(\int_{1}^{*} g(x) d x\) is convergent. (c) Illustrate part (b) by graphing \(f\) and \(g\) on the same screen for \(1 \leqslant x \leq 10 .\) Use your graph to explain intuitively why \(\int_{1}^{\infty} g(x) d x\) is convergent.

\(47-50\) Use integration by parts to prove the reduction formula. $$\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x$$

Use a CAS to find an antiderivative \(F\) of \(f\) such that \(F(0)=0 .\) Graph \(f\) and \(F\) and locate approximately the \(x\) -coordinates of the extreme points and inflection points of \(F .\) $$f(x)=\frac{x^{3}-x}{x^{6}+1}$$