Question

, use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{\pi / 2} \cos ^{2} x \sin x d x $$

Short answer

The integral evaluates to \( \frac{1}{3} \).

Step-by-step solution

Identify Substitution Components

To make the integration easier, identify the substitution. Notice that the integrand involves \( \cos^2 x \sin x \). We can choose \( u = \cos x \), which implies \( du = -\sin x \, dx \). This substitution will make the integral easier to evaluate.

Adjust the Integral with Substitution

Substitute \( u = \cos x \) and adjust the differential and the limits of integration. If \( x = 0 \), then \( u = \cos 0 = 1 \). If \( x = \frac{\pi}{2} \), then \( u = \cos \frac{\pi}{2} = 0 \). Therefore, the integral becomes:\[ -\int_{1}^{0} u^2 \, du \]We change the limits appropriately when substituting the variable.

Simplify the Integral

Since the integral was obtained with reversed boundaries, adjust it by switching the limits and changing the sign. Thus, we have:\[ \int_{0}^{1} u^2 \, du \]

Calculate the Antiderivative

Find the antiderivative of \( u^2 \). The antiderivative is \( \frac{u^3}{3} \). So, the integral becomes:\[ \left[ \frac{u^3}{3} \right]_{0}^{1} \]

Evaluate Definite Integral

Substitute the limits into the antiderivative to evaluate:\[ \left. \frac{u^3}{3} \right|_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \]

Write the Final Result

The evaluated definite integral from \( 0 \) to \( \frac{\pi}{2} \) for \( \cos^2 x \sin x \, dx \) is \( \frac{1}{3} \).

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They represent the accumulation of quantities and allow us to calculate the total change over an interval. In essence, a definite integral computes the area under a curve defined by a function over a specific interval. In our exercise, we are interested in the definite integral of the function \( \cos^2 x \sin x \) over the interval \([0, \pi/2]\).

Here are some key things to remember about definite integrals:

• The limits of integration, denoted by the lower and upper bounds (here, \(0\) and \(\pi/2\)), signify the interval over which the function is integrated.
• The result of a definite integral is a number that represents the net area under the curve between the given bounds.
• Definite integrals can be evaluated using various techniques, including substitution, which is particularly useful when directly integrating a complex function is challenging.

By the end of calculating a definite integral, one typically has a specific value, like \(\frac{1}{3}\) in our problem, representing the total accumulated quantity over an interval.​

Substitution Method

The substitution method is a powerful technique for simplifying integrals. Often viewed as an inverse of the chain rule, substitution helps transform a complicated integral into a simpler one, making it more manageable to evaluate. For our exercise, the substitution method plays a crucial role in converting the integral of \( \cos^2 x \sin x \) into a more workable form.

Here's how substitution works in practice:

• First, choose a substitution that simplifies the integrand. In this case, selecting \( u = \cos x \) helps because it transforms \( \sin x \) into the differential \( du = -\sin x \, dx \).
• After substitution, adjust the limits of integration to reflect changes in the variable. For \( x = 0 \), \( u = 1 \), and for \( x = \pi/2 \), \( u = 0 \).
• Apply the new limits to obtain the integral \(-\int_{1}^{0} u^2 \, du\) and then simplify it by reversing the limits and changing the sign, resulting in \(\int_{0}^{1} u^2 \, du\).

Substitution is especially handy when dealing with products of functions and can often turn a challenging calculation into an easy one with a clear path forward. By simplifying the integrand, it becomes straightforward to proceed to the next step: calculating the antiderivative.

Antiderivative Calculation

The antiderivative calculation is the process of finding a function whose derivative yields the integrand. In our scenario, after simplifying the integral to \( \int_{0}^{1} u^2 \, du \), finding the antiderivative becomes our next focus. The antiderivative of a function reveals the accumulation of the function's values with respect to the variable. Here, we aim to compute the antiderivative of \( u^2 \).

Steps to find an antiderivative:

• Identify the basic form of the function. For \( u^n \), the antiderivative is \( \frac{u^{n+1}}{n+1} \). For our example, the antiderivative of \( u^2 \) is \( \frac{u^3}{3} \).
• After determining the antiderivative, apply the limits of integration. Using the expression \( \left[\frac{u^3}{3}\right]_{0}^{1} \), plug in the upper and lower bounds.
• Evaluate the expression at these bounds: \( \frac{1^3}{3} \) minus \( \frac{0^3}{3} \). This calculation results in the final answer, \( \frac{1}{3} \).

Thus, the process of calculating an antiderivative leads directly to evaluating the definite integral, tying all steps together to reach the conclusion given the function and limits.