Problem 3 - Calculus Early Transcendentals

Question

Describe the method used to integrate \(\sin ^{3} x\).

Step-by-Step Solution

Answer

Question: Find the integral of the function sin^3(x). Answer: The integral of sin^3(x) is -cos(x) + (1/3)cos^3(x) + C, where C is the constant of integration.

Step 1: Rewrite the function using trigonometric identity

Rewrite \(\sin^3 x\) as a product of \(\sin^2 x\) and \(\sin x\). Use the trigonometric identity \(\sin^2 x = 1 - \cos^2 x\) to make the expression easier to work with: $$ \int \sin^3 x dx = \int (\sin^2 x)(\sin x) dx = \int (1 - \cos^2 x)(\sin x) dx. $$

Step 2: Apply substitution

Make the substitution \(u = \cos x\), so that \(du = -\sin x dx\). Replace the variables in the integral: $$ \int (1 - u^2)(-\sin x dx) = -\int (1 - u^2) du. $$

Step 3: Integrate with respect to 'u'

Expand the integrand and find the antiderivative with respect to \(u\): $$ -\int (1 - u^2) du = -\left( \int 1 du - \int u^2 du \right) = -(u - \frac{1}{3}u^3) + C_1, $$ where \(C_1\) is the constant of integration.

Step 4: Replace 'u' with the original variable

Substitute the original variable (\(\cos x\)) back in place of \(u\): $$ -(\cos x - \frac{1}{3}\cos^3 x) + C_1 = -\cos x + \frac{1}{3}\cos^3 x + C. $$ Here, \(C\) is the constant of integration (which may be different from \(C_1\)).

Step 5: Write the final answer

The antiderivative of \(\sin^3 x\) is: $$ \int \sin^3 x dx = -\cos x + \frac{1}{3}\cos^3 x + C. $$

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities are useful in simplifying expressions and solving equations involving trigonometric functions. A commonly used identity is the Pythagorean identity which states that \( \sin^2 x + \cos^2 x = 1 \). This can be rearranged to express \(\sin^2 x\) in terms of \(\cos x\), which is \(\sin^2 x = 1 - \cos^2 x\). In the context of integrals, such identities allow us to convert expressions with higher powers of sine and cosine into forms that are more convenient for integration. For example, integrating \(\sin^3 x\) can be made easier by transforming it into an expression involving \(\cos x\), as seen in our original exercise.

U-Substitution Method

The u-substitution method is a powerful tool for finding antiderivatives of composite functions, similar to the chain rule in differentiation. With u-substitution, we select a part of the integrand and set it equal to \(u\), such that its derivative \(du\) is present as another part of the integrand. This substitution simplifies the integral and often allows us to use basic integration rules. In the exercise's solution, we set \(u = \cos x\) because its derivative \(du = -\sin x dx\) appears in the integral. After substituting \(u\) for \(\cos x\) and \(du\) for \(\sin x dx\), the integral becomes a polynomial in terms of \(u\), which is straightforward to integrate. Remember, the goal is to undo the substitution by replacing \(u\) with the original variable after integrating.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are the inverse operation of taking a derivative. The antiderivative of a function \(f(x)\) is a function \(F(x)\) whose derivative is \(f(x)\), often denoted by \(\int f(x) dx\). Finding an antiderivative means determining the general form of the original function before it was differentiated. This includes a constant of integration, represented as \(C\), since there are infinitely many functions that could differentiate to \(f(x)\). In our exercise, once we've integrated the function with respect to \(u\), we obtain an antiderivative in terms of \(u\), which we then convert back to the original variable to complete the process.

Integration Techniques

Integration techniques involve a variety of methods to evaluate integrals that are not readily solvable with basic integration rules. These techniques include u-substitution, integration by parts, trigonometric substitution, partial fractions, and more. Each technique applies to different kinds of integrals and serves to simplify complex expressions into forms that are easier to integrate. In our problem, we used a combination of trigonometric identities and u-substitution — we first applied a trigonometric identity to express \(\sin^3 x\) in a more integrable form, and then used u-substitution to find the antiderivative efficiently. Choosing the right technique and manipulating the integrand correctly are essential skills in solving integration problems.