§5.3 Integration by substitution (continued)

Summary

This problem centers on finding antiderivatives for the basic trigonometric functions other than $\sin(x)$ and $\cos(x)$. - Consider the indefinite integral $\int \tan(x) \, dx$. By rewriting the integrand as $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and identifying an appropriate function-derivative pair, make a $u$-substitution and hence evaluate $\int \tan(x) \, dx$. - In a similar way, evaluate $\int \cot(x) \, dx$. - Consider the indefinite integral

. Evaluate this integral using the substitution $u = \sec(x) + \tan(x)$. - Simplify the integrand in (c) by factoring the numerator. What is a far simpler way to write the integrand? - Combine your work in (c) and (d) to determine $\int \sec(x) \, dx$. - Using (c)-(e) as a guide, evaluate $\int \csc(x) \, dx$.

Consider the indefinite integral $\int x \sqrt{x-1} \, dx$. - At first glance, this integrand may not seem suited to substitution due to the presence of $x$ in separate locations in the integrand. Nonetheless, using the composite function $\sqrt{x-1}$ as a guide, let $u = x-1$. Determine expressions for both $x$ and $dx$ in terms of $u$. - Convert the given integral in $x$ to a new integral in $u$. - Evaluate the integral in (b) by noting that $\sqrt{u} = u^{1/2}$ and observing that it is now possible to rewrite the integrand in $u$ by expanding through multiplication. - Evaluate each of the integrals $\int x^2 \sqrt{x-1} \, dx$ and $\int x \sqrt{x^2 - 1} \, dx$. Write a paragraph to discuss the similarities among the three indefinite integrals in this problem and the role of substitution and algebraic rearrangement in each.

Consider the indefinite integral $\int \sin^3(x) \, dx$. - Explain why the substitution $u = \sin(x)$ will not work to help evaluate the given integral. - Recall the Fundamental Trigonometric Identity, which states that $\sin^2(x) + \cos^2(x) = 1$. By observing that $\sin^3(x) = \sin(x) \cdot \sin^2(x)$, use the Fundamental Trigonometric Identity to rewrite the integrand as the product of $\sin(x)$ with another function. - Explain why the substitution $u = \cos(x)$ now provides a possible way to evaluate the integral in (b). - Use your work in (a)-(c) to evaluate the indefinite integral $\int \sin^3(x) \, dx$. - Use a similar approach to evaluate $\int \cos^3(x) \, dx$.

For the town of Mathland, MI, residential power consumption has shown certain trends over recent years. Based on data reflecting average usage, engineers at the power company have modeled the town's rate of energy consumption by the function

.

Here, $t$ measures time in hours after midnight on a typical weekday, and $r$ is the rate of consumption in megawatts at time $t$. Units are critical throughout this problem; note that the unit megawatt is itself a rate, which measures energy consumption per unit time. A megawatt-hour is the total amount of energy that is equivalent to a constant stream of 1 megawatt of power being sustained for 1 hour. - Sketch a carefully labeled graph of $r(t)$ on the interval [0,24] and explain its meaning. Why is this a reasonable model of power consumption? - Without calculating its value, explain the meaning of $\int_0^{24} r(t) \, dt$. Include appropriate units on your answer. - Determine the exact amount of energy Mathland consumes in a typical day. - What is Mathland's average rate of power consumption in a given 24-hour period? What are the units on this quantity?