Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integral. \(\int \cos ^{3} x \sin ^{2} x d x\)

Step-by-Step Solution

Answer

The integral of \(\cos ^{3} x \sin ^{2} x d x\) is \(\frac{1}{3}\sin^3 x - \frac{1}{5}\sin^5 x + C\).

Step 1: Power Reducing Formula

We'll use the power reducing formulas for the trigonometric functions \(cos^3(x)\) and \(sin^2(x)\). The general power reducing formulas are \(cos^2(x) = 1 - sin^2(x)\) and \(sin^2(x) = 1 - cos^2(x)\) respectively. Rewriting our integral using those formulas, we get: \(\int (1 - \sin ^{2} x) \sin ^{2} x d x\).

Step 2: Substitution

Let's use a substitution to simplify our integral. We can let \(u = \sin x\), then \(du = \cos x dx\). With that substitution, our integral becomes: \(\int (1 - u^{2}) u^{2} du\).

Step 3: Integrate

Now, let's calculate the integral. \(\int (1 - u^{2}) u^{2} du = \int u^{2} - u^{4} du = \frac{1}{3}u^{2} - \frac{1}{7}u^{5} + C\).

Step 4: Substitute back

The last step is to substitute \(u\) back for \(\sin x\). This gives the following: \(\frac{1}{3}\sin^3 x - \frac{1}{5}\sin^5 x + C\).

Key Concepts

Trigonometric Integrals

Understanding trigonometric integrals is key to mastering calculus. These integrals involve trigonometric functions such as sine, cosine, or tangent, which require special techniques to solve.

One common method for dealing with integrals of the form \(\int \cos^{m} x \sin^{n} x \,dx\) involves simplifying the expression using trigonometric identities. This often includes rewriting functions in terms of \(\sin^{2} x\) and \(\cos^{2} x\). This approach assists in expressing the integral in a more manageable form. For instance, knowing that \(\sin^2(x) = 1 - \cos^2(x)\) can be very useful.

If you have an odd power for either sine or cosine, consider breaking off one sine or cosine to pair with \(dx\) and use substitution techniques to simplify the integral. Acquaintance with these integral properties makes it easier to solve such problems and is a valuable skill for calculus students.

Substitution Method

The substitution method is a powerful tool for simplifying complex integrals. It involves changing variables to make an integral easier to solve, often converting it into a basic form that a student can integrate directly.

For instance, in the integral \(\int \cos^{3} x \sin^{2} x \,dx\), a substitution can simplify the expression tremendously. By setting \(u = \sin x\), the differential \(du = \cos x \,dx\) allows us to convert the original integral into a polynomial in terms of \(u\), specifically \(\int (1-u^2) u^2 \ du\).

This new integral involving \(u\) is simple to evaluate using basic polynomial integration rules. Once the simplified integral is solved, you substitute back the original variable using \(u = \sin x\).

The substitution method is incredibly useful in a variety of calculus problems to simplify the integration process and to handle complex mathematical expressions more effectively.

Power Reducing Formulas

Power reducing formulas are crucial for dealing with integrals involving trigonometric functions raised to a higher power. They help transform complex trigonometric expressions into simpler ones, allowing for easier integration.

These formulas use the identities \(\cos^2 x = 1 - \sin^2 x\) and \(\sin^2 x = 1 - \cos^2 x\). By substituting these into the integral, you reduce the power of trigonometric terms. This simplification often reforms the integral into a format that is straightforward to solve.

Applying power reducing techniques requires recognizing when to use these identities effectively. In our example, substituting \(\cos^3 x\) with \(\cos x (1-\sin^2 x)\) converted a complex trigonometric function into a polynominal form, easing the integration process.

Mastering power reducing formulas enables students to efficiently handle trigonometric integrals, making it simpler to tackle and solve them with confidence in their calculus practice.