Question

Evaluate \(\int \cos ^{3} x \sin ^{2} x d x\)

Short answer

The integral evaluates to \(\frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C\).

Step-by-step solution

Simplify using Trigonometric Identity

First, recognize that you can use trigonometric identities to simplify the expression. Recall the identity: \( \sin^2 x = 1 - \cos^2 x \). Utilize this identity to rewrite part of the integral:\[ \int \cos^3 x \sin^2 x \, dx = \int \cos^3 x (1 - \cos^2 x) \, dx.\]

Expand and Separate the Integral

Expand the expression from Step 1:\[ \int \cos^3 x (1 - \cos^2 x) \, dx = \int (\cos^3 x - \cos^5 x) \, dx.\]Now, you can separate the integral into two parts:\[\int \cos^3 x \, dx - \int \cos^5 x \, dx.\]

Solve Each Integral Separately

Let's address each integral separately.- For \( \int \cos^3 x \, dx \): Use substitution: set \( u = \sin x \), then \( du = \cos x \, dx \). This means \( \cos^2 x \, dx = du \) and \( \cos^3 x = \cos x \cdot \cos^2 x = \cos x \cdot (1 - \sin^2 x) \) becomes \( (1 - u^2)du \). Solve: \( \int (1 - u^2) \, du = \int \, du - \int u^2 \, du \). This integrates to: \( u - \frac{u^3}{3} + C_1 \). Substitute back: \( \sin x - \frac{\sin^3 x}{3} + C_1 \).- For \( \int \cos^5 x \, dx \): Again, use substitution as above, noting \( \cos^5 x = \cos^3 x \cdot \cos^2 x \, dx \). With \( u = \sin x \), transforms \( \cos^3 x \, dx = (1-u^2) \, du \) and \( \cos^5 x = \cos^3 x \cdot (1-u^2) \) becomes \( (1-u^2)(1-u^2)du \). Solve: \( \int (1 - 2u^2 + u^4) \, du = u - \frac{2u^3}{3} + \frac{u^5}{5} + C_2 \). Substitute back: \( \sin x - \frac{2 \sin^3 x}{3} + \frac{\sin^5 x}{5} + C_2 \).

Combine and Simplify the Result

Combine the results from each integral:\( \int \cos^3 x \, dx = \sin x - \frac{\sin^3 x}{3} \)\( \int \cos^5 x \, dx = \sin x - \frac{2 \sin^3 x}{3} + \frac{\sin^5 x}{5} \)Subtract them:\[ (\sin x - \frac{\sin^3 x}{3}) - (\sin x - \frac{2 \sin^3 x}{3} + \frac{\sin^5 x}{5}) \]Simplify to get:\[ \frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C \]Thus, the integral evaluates to:\[ \int \cos^3 x \sin^2 x \ dx = \frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C\]

Conclusion: Evaluate and Finalize

The original integral has been evaluated as:\[ \int \cos^3 x \sin^2 x \, dx = \frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C.\]Make sure to include the constant of integration \( C \) to account for the indefinite integral.

Key concepts

Trigonometric Integration

Trigonometric integration is a technique used to evaluate integrals involving trigonometric functions such as sine (\( \sin x \)) and cosine (\( \cos x \)).
This method is especially useful when the integrals involve powers of these functions or products of different trigonometric functions.
In our problem, we encounter an integral involving \( \cos^3 x \sin^2 x \).

By using trigonometric identities, we can transform the integral to a more manageable form. A key identity here is \( \sin^2 x = 1 - \cos^2 x \).
Employing such identities allows you to manipulate and simplify expressions, making the integration process easier.
Always examine if such identities can simplify integrals further or if different identities can be applied in succession for a better simplification.

Substitution Method

The substitution method is a powerful tool for solving integrals where a direct approach seems complex.
In our problem, we simplify the expression by setting a substitution such as \( u = \sin x \), which translates parts of the function into a form that's easier to integrate.

The primary idea is to change the variable of integration to another variable, which can lead to a function that is easier to handle.
Substituting also involves finding the derivative replacement, in this case \( du = \cos x \, dx \).
This substitution transforms the integral into more familiar and simpler integrals, which can then be solved using basic integration rules.

• Choose a substitution that simplifies the integrand.
• Express \( dx \) in terms of \( du \).
• Change limits if definite integration, otherwise back-substitute when integration is complete.

Through substitution, we transform the integral into a solvable form by integrating \( (1 - u^2) \) and using the power rule for integration.

Indefinite Integrals

Indefinite integrals refer to finding functions that, when differentiated, yield the integrand.
An indefinite integral includes a constant of integration (\( C \)) because both constants and functions can have the same derivatives.
In integral calculus, finding indefinite integrals translates to finding an antiderivative of the given function.

When dealing with trigonometric functions like \( \int \cos^3 x \sin^2 x \, dx \), evaluating step-by-step simplifies the process.
Each outcome reflects multiple possible functions differing by constants.

• Integrate terms one at a time.
• Be sure to include the constant of integration once you reach the solution.

In our solution, we carefully solve for each part separately, and at the end, combine and simplify these results before adding the constant \( C \) to represent all possible solutions.