Problem 32 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int \sin ^{2} x \cos x d x$$

Step-by-Step Solution

Answer

The integral is \( \frac{\sin^3 x}{3} + C \). Verification by differentiation confirms it is correct.

Step 1: Identify the Integration Technique

Notice that the integral \( \int \sin^2 x \cos x \, dx \) can be solved using a substitution method. We can apply a trigonometric identity or a direct u-substitution method.

Step 2: Choose a Substitution

We choose the substitution method. Let \( u = \sin x \), then \( du = \cos x \, dx \). This substitution transforms the integral into \( \int u^2 \, du \).

Step 3: Perform the Integration

Now integrate \( \int u^2 \, du \). The integral of \( u^n \) is \( \frac{u^{n+1}}{n+1} + C \). Applying this, we get \( \frac{u^3}{3} + C \).

Step 4: Substitute Back

Replace \( u \) back with \( \sin x \). Thus, the integral becomes \( \frac{(\sin x)^3}{3} + C \), or \( \frac{\sin^3 x}{3} + C \).

Step 5: Differentiate to Verify

Differentiate \( \frac{\sin^3 x}{3} + C \) to ensure correctness. Use the chain rule: \( \frac{d}{dx}[\sin^3 x] = 3\sin^2 x \cdot \cos x \). Dividing by 3 gives \( \sin^2 x \cos x \), confirming the integral matches the original function.

Key Concepts

u-substitution method

The u-substitution method is a technique used to simplify the integration process by making a substitution that turns a complicated integral into a simpler one. This method is particularly useful when dealing with integrals that involve compositions of functions, such as powers of trigonometric functions or algebraic functions composed with other expressions. When approaching an integral using u-substitution, the key steps include:

• Identifying a portion of the integral that can be replaced with a single variable, typically denoted by \( u \).
• Calculating \( du \), the differential of \( u \), to replace the corresponding part of the integrand.
• Substituting \( u \) and \( du \) into the integral, thus transforming it into a new integral that is easier to evaluate.

For example, in \( \int \sin^2 x \cos x \, dx \), we let \( u = \sin x \). This makes \( du = \cos x \, dx \), transforming the integral into \( \int u^2 \, du \). The resulting integral is basic to solve, yielding \( \frac{u^3}{3} + C \). Finally, substitute back \( u = \sin x \) to get the original variable terms in the solution.

trigonometric substitution

Trigonometric substitution is a technique used to evaluate integrals involving square roots of quadratic expressions by substituting trigonometric identities. This method simplifies the integral by taking advantage of the Pythagorean identities, which relate trigonometric functions to one another. Commonly used identities include:

• \( 1 - \sin^2 x = \cos^2 x \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( \sec^2 x - 1 = \tan^2 x \)

Trigonometric substitution is particularly effective for integrals involving expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \). For instance, to solve \( \int \sqrt{1 - x^2} \, dx \), a common substitution is \( x = \sin \theta \) or \( x = \cos \theta \), taking advantage of the symmetry and simplicity of trigonometric functions. While this method wasn't needed in the given problem \( \int \sin^2 x \cos x \, dx \), similar principles could guide choosing substitutions or guiding simplifications.

differentiation verification

Differentiation verification is a process of checking the correctness of an integration result by differentiating it. This ensures that the indefinite integral found is indeed a function whose derivative matches the original integrand.Steps for verification include:

• Differentiate the obtained integral result.
• Simplify the resulting expression, ensuring it is equivalent to the original integrand.

In our example, the calculated integral was \( \frac{\sin^3 x}{3} + C \). Differentiating this expression involves using the chain rule. The derivative \( \frac{d}{dx}[\sin^3 x] \) is given by \( 3\sin^2 x \cdot \cos x \). Dividing by 3 simplifies to \( \sin^2 x \cos x \), precisely matching the original integrand.This process of differentiation acts as a reliable verification tool, confirming the accuracy of our integration method and result. It ensures that the solution to the calculus exercise is both valid and precise.