§5.3 Integration by substitution (continued)

Summary

Rewriting $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and substituting $u = \cos(x)$ shows that $\int\tan(x)\,dx$ equals

For $\int\sin^3(x)\,dx$, the substitution $u = \sin(x)$ fails because

Writing $\sin^3(x) = \sin(x)(1 - \cos^2(x))$ and using $u = \cos(x)$ gives $\int\sin^3(x)\,dx$ equal to

A town consumes power at rate $r(t) = 4 + \sin(0.263t + 4.7) + \cos(0.526t+9.4)$ megawatts. The integral $\int_0^{24} r(t)\,dt$

Using the antiderivative $\frac{1}{3}\cos^3(x) - \cos(x) + C$, evaluate $\int_0^{\pi/2}\sin^3(x)\,dx$ as a fraction.

Using $\int x\sqrt{x-1}\,dx = \frac{2}{5}(x-1)^{5/2} + \frac{2}{3}(x-1)^{3/2} + C$, evaluate $\int_1^2 x\sqrt{x-1}\,dx$ as a fraction.

Using $\int_0^{24} r(t)\,dt \approx 95.7809$ megawatt hours, compute the average power draw over the day, rounded to two decimals.

The integral $\int\cot(x)\,dx$ equals $-\ln|\csc(x)| + C$.