91Ó°ÊÓ

The study of sawtooth waves in electrical engineering leads to integrals of the form $$\int_{-\pi / \omega}^{\pi / \omega} t \sin (k \omega t) d t$$ where \(k\) is an integer and \(\omega\) is a nonzero constant. Evaluate the integral.

there is a good chance that your CAS will not be able to evaluate the integral 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. $$\int_{0}^{1 / 4} \sec \pi x \tan \pi x d x$$

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

(a) Complete the square, make an appropriate \(u\) -substitution, and then use the Endpaper 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). $$\int \frac{1}{x^{2}+4 x-5} d x$$

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

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

there is a good chance that your CAS will not be able to evaluate the integral as stated. If this is so, make a substitution that converts the integral into one that your CAS can evaluate. $$\int_{0}^{1} 3^{x} \sqrt{9^{x}-1} d x$$

(a) Complete the square, make an appropriate \(u\) -substitution, and then use the Endpaper 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). $$\int \sqrt{3-2 x-x^{2}} d x$$

(a) Complete the square, make an appropriate \(u\) -substitution, and then use the Endpaper 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). $$\int \frac{x}{\sqrt{5+4 x-x^{2}}} d x$$