Problem 6 - Calculus: One and Several Variables

Question

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sin ^{3} x \cos ^{2} x d x$$.

Step-by-Step Solution

Answer

The short answer to the given integral is \(\int \sin^{3}{x}\cos^{2}{x} dx = \frac{1}{4}\sin^{4}{x} - \frac{1}{6}\sin^{6}{x} + C\).

Step 1: Express the integrand in terms of sine and cosine functions

We are given the integral: \(\( \int \sin^{3} x \cos^{2} x dx \)\)

Step 2: Use substitution to simplify the integral

Let us use the substitution: \(\( u = \sin{x} \)\). Then, \(\( du = \cos{x} dx \)\). Before we use this substitution, let's rewrite the integral in terms of sine and cosine: \(\( \int (\sin^{2}{x} \cdot \sin{x}) (\cos{x} \cdot \cos{x}) dx \)\) Now, we can use the substitution. \(\( \int (u^{2} \cdot u) (1 - u^{2}) du \)\) Simplify the integrand: \(\( \int (u^{3} - u^{5}) du \)\)

Step 3: Evaluate the integral using integration rules

Now, we can integrate the function term by term: \(\( \int u^{3} du - \int u^{5} du \)\) \(\( \frac{1}{4}u^{4} - \frac{1}{6}u^{6} + C \)\) Now, to express the result in terms of x, substitute back the initial expression for u: \(\( \frac{1}{4}\sin^{4}{x} - \frac{1}{6}\sin^{6}{x} + C \)\) Therefore, the solution to the given integral is: \(\( \int \sin^{3}{x}\cos^{2}{x} dx = \frac{1}{4}\sin^{4}{x} - \frac{1}{6}\sin^{6}{x} + C \)\).

Key Concepts

Integration by Substitution

Integration by substitution is a technique that helps to simplify an integral by changing variables. This method is especially useful when dealing with complicated expressions, like products of trigonometric functions. In our exercise, we have an integral of the form:

• \( \int \sin^{3} x \cos^{2} x \, dx \)

The goal is to make this integral easier to handle. Here, we choose a new variable \( u \) such that \( u = \sin{x} \). This transforms our differential as well, with \( du = \cos{x} \, dx \). By substituting, the integral becomes:

• \( \int (u^{2} \cdot u) (1 - u^{2}) \, du \)

This substitution transforms the complex trigonometric functions into a simple polynomial, making the integration process straightforward. Always remember to revert back to the original variable at the end of the solution by substituting \( u \) back with \( \sin{x} \).

Trigonometric Identities

Trigonometric identities are crucial in simplifying expressions before integration or other operations. They help reframe complex trigonometric expressions into simpler forms. For the given problem, the power-reducing identity comes into play:

• \( \cos^2{x} = 1 - \sin^2{x} \)

This identity allows us to express \( \cos^2{x} \) in terms of \( \sin{x} \), facilitating the substitution process. In the context of our integral, the identity modifies the integrand from \( \cos^{2}{x} \) into \( 1 - \sin^2{x} \).This kind of transformation is an essential technique. It reduces the complexity of manual calculations or when performing substitutions for integration. Memorizing key trigonometric identities will greatly aid in solving such integral problems efficiently.

Definite Integrals

Although our original problem is an indefinite integral, it's vital to link this to the concept of definite integrals since they are often used together. A definite integral evaluates the accumulation of quantities, providing a numerical value by incorporating limits of integration.For example, if we adapted our integral to:

• \( \int_{a}^{b} \sin^{3}{x}\cos^{2}{x} \, dx \)

Here, \([a, b]\) are the limits of integration. The result of evaluating a definite integral gives the area under the curve of the function between \( x = a \) and \( x = b \). This is achieved by computing the antiderivative and substituting the limits:

• Calculate the antiderivative as before.
• Evaluate \( F(b) - F(a) \).

Understanding how to apply these techniques allows students to solve a wide range of integration problems more thoroughly.