Problem 50 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{-\pi / 6}^{\pi / 6} \sin ^{2} x \cos x d x $$

Step-by-Step Solution

Answer

The evaluated integral is \( \frac{1}{12} \).

Step 1: Identify the Substitution

We begin by choosing an appropriate substitution to simplify the integral. Since the integrand is \( \sin^2(x) \cos(x) \), let us set \( u = \sin(x) \). Then, the derivative \( du = \cos(x) \, dx \), allowing us to substitute \( \cos(x) \, dx \) with \( du \).

Step 2: Change the Limits of Integration

We need to adjust the limits of integration based on our substitution. Initially, the limits for \( x \) are from \( -\pi/6 \) to \( \pi/6 \). Under the substitution \( u = \sin(x) \), when \( x = -\pi/6 \), \( u = \sin(-\pi/6) = -1/2 \), and when \( x = \pi/6 \), \( u = \sin(\pi/6) = 1/2 \). Therefore, the new limits for \( u \) are from \(-1/2\) to \(1/2\).

Step 3: Rewriting the Integral

With the substitution, the integral \( \int_{-rac{\pi}{6}}^{\frac{\pi}{6}} \sin^2(x) \cos(x) \, dx \) becomes \( \int_{-1/2}^{1/2} u^2 \, du \). This is because \( \sin^2(x) \) becomes \( u^2 \), and \( \cos(x) \, dx \) is replaced by \( du \).

Step 4: Evaluate the Definite Integral

Evaluate the new integral \( \int_{-1/2}^{1/2} u^2 \, du \). The antiderivative of \( u^2 \) is \( \frac{u^3}{3} \). Thus, \[\int u^2 \, du = \left[ \frac{u^3}{3} \right]_{-1/2}^{1/2}.\]Evaluate this by plugging in the bounds:\[\left. \frac{u^3}{3} \right|_{-1/2}^{1/2} = \frac{(1/2)^3}{3} - \frac{(-1/2)^3}{3}.\]This simplifies to \( \frac{1}{3} [ \frac{1}{8} - (-\frac{1}{8})] = \frac{1}{3} \times \frac{2}{8} = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} \).

Step 5: Conclusion

The evaluated definite integral is \( \frac{1}{12} \). This is the final result after substitution and integration.

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique in calculus, ideal for making complex integrals manageable. The basic idea is to transform the integrand into a simpler form by changing variables. Let's explore how this method works through the try-out of a typical integral.When approaching an integral like \( \int \sin^2(x) \cos(x) \, dx \), we can simplify this using substitution. Suppose we set \( u = \sin(x) \). This choice is strategic because the derivative of \( \sin(x) \) is \( \cos(x) \), which directly appears in the integrand. Therefore, \( du = \cos(x) \, dx \), allowing us to replace \( \cos(x) \, dx \) in the integral with \( du \).

• Choose a substitution: Set \( u = \sin(x) \).
• Find the derivative: \( du = \cos(x) \, dx \).
• Rewrite the integral: Replace terms to get \( \int u^2 \, du \).

This substitution simplifies our calculation significantly: after changing the variable, the integral features \( u^2 \), a much easier expression to evaluate than the original \( \sin^2(x) \cos(x) \). Remember, the goal of substitution is to transform the integral into a form you can solve more easily.

Trigonometric Integrals

Trigonometric integrals involve functions like sine and cosine, and often appear in problems that need special techniques to simplify. In our exercise, the trigonometric integral \( \int \sin^2(x) \cos(x) \, dx \) can become simpler when employing trigonometric identities and substitutions.To tackle trigonometric integrals:

• Consider possible substitutions. Often, using identities like \( \sin(2x) = 2 \sin(x) \cos(x) \) can guide your choices.
• For \( \sin^2(x) \) and \( \cos^2(x) \), use the identity: \( \sin^2(x) + \cos^2(x) = 1 \) to simplify the integral.
• Identify terms that can be represented as derivatives of simpler trigonometric functions.

In our exercise's case, the choice of \( \sin(x) \) for substitution works well since the derivative, \( \cos(x) \), is conveniently present in the integrand. Often in trigonometric integrals, the success hinges on recognizing these overlapping structures, allowing for easier integration.

Calculus Problem Solving

Successfully solving calculus problems, like evaluating definite integrals, involves understanding both theory and practical strategies. Here, we solve the integral \( \int_{-\pi/6}^{\pi/6} \sin^2(x) \cos(x) \, dx \) using substitution, demonstrating clear problem-solving techniques.Steps to effective problem solving in calculus:

• Understand the problem: Carefully analyze the structure of the integral and identify the functions involved.
• Plan your approach: Decide on a strategy, such as substitution, by considering potential simplifications.
• Execute the plan: Carry out substitutions, revert limits if necessary, and systematically manipulate the integral.
• Evaluate and Reflect: Substitute back any variables you've introduced and evaluate with revised limits. Reflect on the steps to ensure accuracy.

In our example, effective problem solving transformed a complex-looking trigonometric integral into a straightforward polynomial form by using substitution. Integration becomes trivial thereafter, with the antiderivative calculation and evaluation yielding the final answer, \( \frac{1}{12} \). Understanding this structured process enhances both confidence and capability in tackling varied calculus problems.