Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integral. \(\int \cos x \sin ^{2} x d x\)

Step-by-Step Solution

Answer

The solution to the integral \(\int \cos(x) \sin^2(x) dx\) is \(-(1/2)\cos^2(x) + (1/4)\cos^4(x) + C\)

Step 1: Trigonometric identity and substitution setup

Use the trigonometric identity \(\sin^2(x) = 1-\cos^2(x)\). Then, let \(u = \cos(x)\), then \(du = -\sin(x)dx\). So, the integral expression becomes: \(-\int u (1-u^2) du\).

Step 2: Evaluate the integral

Evaluate the integral \(-\int u (1 - u^2) du\). Use the power rule of integral \(\int x^n dx = (1/n+1)x^{n+1} + C\), to integrate \(u\) and \(u^3\). Hence, \(-\int u (1-u^2) du = -[ (1/2)u^2 - (1/4)u^4 ] + C\).

Step 3: Back-substitution

Substitute \(u\) back as \(\cos(x)\). So the final answer to the problem would be: \(-(1/2)\cos^2(x) + (1/4)\cos^4(x) + C\) .

Key Concepts

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for every value of the angles involved. They are often used to simplify expressions or solve integrals. In the context of the given exercise, we used the identity \(\sin^2(x) = 1 - \cos^2(x)\).
This particular identity arises from the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\). By rearranging this, we express \(\sin^2(x)\) as \(1 - \cos^2(x)\).
Using trigonometric identities is like using a mathematical trick to change the form of your integral into something easier to work with. When you see \(\sin^2(x)\), it's often helpful to rewrite it using a trigonometric identity if it facilitates integration.

Substitution Method

The substitution method, sometimes called "u-substitution," is an integration technique similar to the reverse of the chain rule in differentiation. It involves substituting a part of the integral with a new variable to simplify the expression.
In our exercise, we used \(u = \cos(x)\) and found \(du = -\sin(x) dx\). This substitution simplifies our integral, making it easier to solve.

• First, identify a part of the integral that, when substituted, will reduce the complexity of the integral. Here \(\cos(x)\) was chosen.
• Next, express \(dx\) in terms of \(du\), accounting for any changes like the negative sign in \(du = -\sin(x) dx\).
• The goal is to rewrite the entire integral in terms of \(u\) and \(du\).

After substitution, your integral is transformed into a new integral that is (hopefully) easier to evaluate.

Power Rule of Integration

The power rule for integration is one of the fundamental rules of calculus. It states that the integral of \(x^n\) is \((1/(n+1)) \cdot x^{n+1} + C\) where \(n eq -1\). This rule provides a straightforward way to integrate powers of variables.
In our solution, we applied the power rule to each term separately:

• For \(u^2\), use \(n=2\) to get \(-\frac{1}{2}u^2\).
• For \(u^4\), use \(n=4\) to get \(\frac{1}{4}u^4\).

The result of these integrals is a sum of simpler terms. We then back-substitute the original variable to complete the process and express the final result in terms of the initial variable \(x\). Remember that \(C\) represents the constant of integration, accounting for all constants that vanish during differentiation.