91Ó°ÊÓ

Let \(f(x)=\cos \left(x-x^{2}\right)\) (a) Use a CAS to approximate the maximum value of \(\left|f^{(4)}(x)\right|\) on the interval [0,1] (b) How large must the value of \(n\) be in the approximation \(S_{n}\) of \(\int_{0}^{1} f(x) d x\) by Simpson's rule to ensure that the absolute error is less than \(10^{-4} ?\) (c) Estimate the integral using Simpson's rule approximation \(S_{n}\) with the value of \(n\) obtained in part (b).

(a) Complete the square, make an appropriate \(u\) -substitution, and then use the End paper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution or square completion), and then confirm that the result is equivalent to that in part (a). $$\int \frac{1}{x^{2}+6 x-7} d x$$

Show that $$\int_{0}^{1} \frac{x}{x^{4}+1} d x=\frac{\pi}{8}$$

Find the area of the region between the \(x\) -axis and the curve \(y=e^{-3 x}\) for \(x \geq 0\).

Evaluate the integral. $$\int \cot ^{3} x \csc ^{3} x d x$$

There is a good chance that your CAS will not be able to evaluate these integrals as stated. If this is so, make a substitution that converts the integral into one that your CAS can evaluate. $$\int \cos x \sin x \sqrt{1-\sin ^{4} x} d x$$

Use partial fractions to derive the integration formula $$\int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+C$$

There is a good chance that your CAS will not be able to evaluate these integrals as stated. If this is so, make a substitution that converts the integral into one that your CAS can evaluate. $$\int(x \cos x+\sin x) \sqrt{1+x^{2} \sin ^{2} x} d x$$

Evaluate the integral using tabular integration by parts. $$\int x^{3} \sqrt{2 x+1} d x$$

Evaluate the integral. $$\int \cot ^{2} 3 t \sec 3 t d t$$