Problem 35 - Biocalculus Calculus for the Life Sciences

Question

\(35-36\) Verify by differentiation that the formula is correct. \(\int \cos ^{3} x d x=\sin x-\frac{1}{3} \sin ^{3} x+C\)

Step-by-Step Solution

Answer

The given integral formula is correct.

Step 1: Differentiate the Right Side

Differentiate the expression on the right-hand side of the given formula. We perform the differentiation of each term separately. Start with the differentiation of \( \sin x \), which is \( \cos x \). Then, for the term \( -\frac{1}{3} \sin^3 x \), use the chain rule. The derivative is \(-\frac{1}{3} (3 \sin^2 x) \cos x = -\sin^2 x \cos x\) since the derivative of \( \sin x \) is \( \cos x \). The derivative of the constant \( C \) is zero.

Step 2: Simplify the Result

Combine like terms from the differentiated expression: \( \cos x - \sin^2 x \cos x \). This expression can be rewritten using the identity \( \sin^2 x = 1 - \cos^2 x \). Substitute to get \( \cos x - (1 - \cos^2 x) \cos x = \cos^3 x \). Therefore, the simplified differentiation result is \( \cos^3 x \).

Step 3: Verification by Comparison

Compare the simplified result \( \cos^3 x \) from differentiation with the original integrand on the left-hand side \( \cos^3 x \). Since they match, it confirms that the differentiation of the given integral's right-hand side yields the original integrand, verifying the correctness of the formula.

Key Concepts

Differentiation

Differentiation is the process of calculating the derivative of a function. It measures how a function changes as its input changes. In integral calculus, differentiation is often used to verify the correctness of antiderivatives. In the given problem, we verify by differentiating the expression to ensure it returns to the original integrand.

• The derivative of a basic function like \( \sin x \) is straightforward; it simply becomes \( \cos x \).
• When differentiating more complex terms, such as \( -\frac{1}{3} \sin^3 x \), it's essential to remember that these are composite functions which require the use of differentiation techniques such as the chain rule.
• Constant terms like \( C \) disappear upon differentiation as their rate of change is zero.

By carefully differentiating each component and summing the results, we ensure our differentiated expression is accurate.

Integration

Integration is the reverse process of differentiation. It's like reconstructing the original function from its derivative. In this exercise, we were provided with a formula for the integral of \( \cos^3 x \). To assure us of its accuracy, we differentiate the result to see if it returns us to the original integrand.Understanding integration involves:

• Finding an antiderivative, which is a function whose derivative gives the integrand you started with.
• Including the constant of integration \( C \), which represents any constant added to the indefinite integral.

By verifying through differentiation, the integrity of the integration formula is confirmed.

Chain Rule

The chain rule is a fundamental technique in calculus used when differentiating composite functions. It works by breaking down these functions into simpler parts to make differentiation possible. In this exercise, the term \( -\frac{1}{3} \sin^3 x \) required the application of the chain rule.Here's a quick guide on how the chain rule was used:

• The outer function is \( -\frac{1}{3} (u)^3 \), where \( u = \sin x \).
• Differentiate the outer function: \(-\frac{1}{3} \times 3u^2 = -u^2 \).
• Next, differentiate the inner function \( u = \sin x \), which gives \( \cos x \).
• Multiply the derivatives: \( -\sin^2 x \times \cos x \).

By using the chain rule properly, we ensure the derivative is computed accurately, allowing us to confirm the integration result.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables. They are crucial tools in simplifying expressions or solving equations in calculus. In this exercise, the key identity used is \( \sin^2 x = 1 - \cos^2 x \). This identity helped simplify the differentiated expression.Why trigonometric identities matter:

• They make complex trigonometric expressions easier to work with.
• Simplifying expressions can reveal equivalent yet simpler forms.
• They can aid in comparing and substituting expressions to verify calculus problems like this one.

By applying trigonometric identities, we reduce the expression to \( \cos^3 x \), confirming our differentiated result matches the original integrand.