Problem 15 - Applied Calculus

Question

Find each integral. $$ \int \sin ^{2} t \cos t d t $$

Step-by-Step Solution

Answer

The integral is \( \frac{\sin^3 t}{3} + C \).

Step 1: Identify the Integration Method

The integral \( \int \sin^{2} t \cos t \, dt \) can be solved using substitution. We will use the substitution method to simplify the integrand.

Step 2: Choose a Substitution Variable

Let \( u = \sin t \). Then, differentiate \( u \) with respect to \( t \), yielding \( \frac{du}{dt} = \cos t \). This implies \( du = \cos t \, dt \). The integral then changes into \( \int u^2 \, du \).

Step 3: Integrate with Respect to New Variable

Now integrate \( \int u^2 \, du \). The power rule for integration tells us that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Thus, \( \int u^2 \, du = \frac{u^3}{3} + C \), where \( C \) is the integration constant.

Step 4: Substitute Back to Original Variable

Replace \( u \) with \( \sin t \) to convert back into the original variable. This gives us \( \frac{(\sin t)^3}{3} + C \) or \( \frac{\sin^3 t}{3} + C \).

Step 5: Final Answer Verification

Verify by differentiating the result to ensure it equals the original integrand. Differentiating \( \frac{\sin^3 t}{3} + C \) yields \( \sin^2 t \cos t \), which matches the original integrand.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in calculus for simplifying the integration process. In essence, it involves finding a substitution that makes the integrand easier to handle.
The method typically follows these steps:

• Identify a part of the integrand that can be replaced with a simpler variable, usually denoted as \( u \).
• Differentiate \( u \) with respect to the original variable, and solve for \( du \) in terms of the original differential.
• Substitute \( u \) and \( du \) into the integral, resulting in a simpler integral, typically in terms of \( u \).
• Complete the integration in this new form.
• Finally, substitute back the original variable in place of \( u \).

In the example from the original exercise, setting \( u = \sin t \) transforms \( \ \int \sin^{2} t \cos t \, dt \ \) into \( \ \int u^2 \, du \ \), which is straightforward to solve.

Integration by Parts

Integration by parts is another essential technique for evaluating integrals, particularly useful when the integrand is a product of two functions. It is derived from the product rule of differentiation and is represented by:

• \( \ \int u \, dv = uv - \int v \, du \ \)

Here, you need to choose parts of the integrand as \( u \) and \( dv \). The method involves:

• Choosing \( u \) and \( dv \).
• Finding \( du \) by differentiating \( u \), and \( v \) by integrating \( dv \).
• Plugging these into the integration by parts formula.

Although we didn't apply this technique in the given exercise, it's crucial when substitution doesn't simplify the integral sufficiently.

Indefinite Integrals

Indefinite integrals are antiderivatives of functions, representing a family of functions whose derivative is the integrand. The result of an indefinite integral is always expressed with a constant of integration, \( C \), written as:

• \( \ \int f(x) \, dx = F(x) + C \ \)

This process involves finding a function whose derivative gives back the original function within the integrand.
The indefinite integral is fundamental to understanding how functions evolve over inputs.
In the original exercise, after substitution and integration steps, the result is an indefinite integral expressed as \( \ \frac{\sin^3 t}{3} + C \ \).
Verifying by differentiation confirms the accuracy of antiderivatives, linking back to the original integrand \( \ \sin^2 t \cos t \ \).