Problem 52 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \( C = 0 \)). \( \displaystyle \int \sin x \cos^4 x \, dx \)

Step-by-Step Solution

Answer

The antiderivative is \( -\frac{(\cos x)^5}{5} \).

Step 1: Choose a Substitution

We will use substitution to solve this integral. Choose \( u = \cos x \). Consequently, the derivative will be \( du = -\sin x \,dx \) or \(-du = \sin x \, dx\).

Step 2: Rewrite the Integral

Substitute \( u \) into the integral:\[ \int \sin x \cos^4 x \, dx = \int -u^4 \, du \] The integral simplifies to:\[ -\int u^4 \, du \].

Step 3: Integrate

Perform the integration with respect to \( u \):\[ -\int u^4 \, du = -\frac{u^5}{5} + C \].

Step 4: Reverse the Substitution

Replace \( u \) with \( \cos x \):\[ -\frac{(\cos x)^5}{5} + C \].

Step 5: Apply the Constant of Integration

Since \( C = 0 \) is given, the antiderivative is:\[ -\frac{(\cos x)^5}{5} \].

Step 6: Verify by Differentiation

Differentiate \( -\frac{(\cos x)^5}{5} \) to verify:\[ \frac{d}{dx}\left(-\frac{(\cos x)^5}{5}\right) = (\cos x)^4 \sin x \], which confirms the original integrand.

Step 7: Graph the Functions

Graph both \( \sin x \cos^4 x \) and \( -\frac{(\cos x)^5}{5} \) to ensure the antiderivative has the correct shape corresponding to the slope of the initial function.

Key Concepts

Substitution Method

When tackling complex integrals, the substitution method is a valuable tool. It involves substituting part of the integral with a new variable, which simplifies the problem. For this exercise, we chose the substitution \( u = \cos x \). This choice of \( u \) is strategic because it allows us to rewrite the original integral in terms of \( u \), making it easier to integrate.

Here's how to execute it:

• Choose \( u = \cos x \), which implies \( du = -\sin x \, dx \).
• Then, express \( \sin x \, dx \) as \( -du \), and substitute it into the integral.

This results in the integral \( \int -u^4 \, du \), a much simpler form. It's a technique that eliminates the complexity of the original integrand.

Antiderivative

Finding the antiderivative is the key to solving indefinite integrals. An antiderivative of a function \( f \) is another function \( F \) whose derivative equals \( f \). For this exercise, after our substitution, we arrived at the integral \( \int -u^4 \, du \).

To integrate this, we apply the power rule for integration, which states:

• If \( n eq -1 \), then \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \).

Applying this rule gives us \( -\frac{u^5}{5} + C \). Once we reverse the substitution by substituting back \( u = \cos x \), the antiderivative becomes \( -\frac{(\cos x)^5}{5} + C \). Since \( C = 0 \) is specified, the final result for the antiderivative is \( -\frac{(\cos x)^5}{5} \).

Differentiation Verification

Verification through differentiation is an essential step to confirm the correctness of an antiderivative. It involves taking the derivative of the found antiderivative and checking if it matches the original function under the integral.

For our result, which is \( -\frac{(\cos x)^5}{5} \), we compute its derivative:

• First, apply the power rule and chain rule to differentiate.
• The result should be \( (\cos x)^4 \sin x \).

This indeed matches the integrand \( \sin x \cos^4 x \), confirming that our antiderivative is correct. Differentiation verification is crucial as it serves as a reliable check ensuring you did not make mistakes during integration.

Graphing Functions

Graphing functions is a visual method to verify and understand the behavior of a function and its antiderivative. By plotting both the original function \( \sin x \cos^4 x \) and its antiderivative \( -\frac{(\cos x)^5}{5} \), we can visually affirm the relationship between them.

Here's how graphing helps:

• The curve of the antiderivative should represent the accumulation of the area under the curve of the original function.
• The slopes of the antiderivative should correspond to the values of the original function.

By ensuring the graphs align in this manner, you gain an additional layer of confidence in your calculations. It's an intuitive method to visualize the accuracy and nature of integration results, fostering a deeper understanding of the concepts.