91Ó°ÊÓ

Sketch the graph of a continuous function on \([0,2]\) for which the Trapezoidal Rule with \(n=2\) is more accurate than the Midpoint Rule.

Evaluate the integral. $$\int \sin 8 x \cos 5 x d x$$

(a) Use the table of integrals to evaluate \(F(x)=\int f(x) d x\) where $$f(x)=\frac{1}{x \sqrt{1-x^{2}}}$$ What is the domain of \(f\) and \(F ?\) (b) Use a CAS to evaluate \(F(x) .\) What is the domain of the function \(F\) that the CAS produces? Is there a discrepancy between this domain and the domain of the function \(F\) that you found in part (a)?

Computer algebra systems sometimes need a helping hand from human beings. Try to evaluate $$\int(1+\ln x) \sqrt{1+(x \ln x)^{2}} d x$$ with a computer algebra system. If it doesn't return an answer, make a substitution that changes the integral into one that the CAS can evaluate.

Evaluate the integral. $$\int \cos \pi x \cos 4 \pi x d x$$ x d x$$

(a) Prove the reduction formula $$\int \cos ^{n} x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x$$ (b) Use part (a) to evaluate \(\int \cos ^{2} x d x\) . (c) Use parts (a) and (b) to evaluate \int \(\cos ^{4} x d x\) .

\(41-46\) Sketch the region and find its area (if the area is finite). $$S=\\{(x, y) | x \geq 0,0 \leqq y \leqslant x /\left(x^{2}+9\right)\\}$$

Sketch the graph of a continuous function on \([0,2]\) for which the right endpoint approximation with \(n=2\) is more accurate than Simpson's Rule.

\(\begin{array}{l}{39-50 \text { Make a substitution to express the integrand as a rational }} \\ {\text { function and then evaluate the integral. }}\end{array}\) $$ \int_{1 / 3}^{3} \frac{\sqrt{x}}{x^{2}+x} d x $$

\(1-80\) Evaluate the integral. $$\int \sqrt{1+e^{x}} d x$$