Question

Evaluate the given integral. $$ \int \sin ^{3} x \cos x d x $$

Short answer

\( \frac{\sin^4 x}{4} + C \)

Step-by-step solution

Identify the integral

The given integral is \( \int \sin^3 x \cos x \, dx \). This is an integral involving a power of sine and a cosine term, which suggests the use of a substitution method could be appropriate.

Choose a substitution

Select \( u = \sin x \). This substitution is selected because \( \sin x \) has an entire power of 3, and its derivative, \( \cos x \, dx = du \), also appears in the integral. This makes the substitution straightforward.

Perform the substitution

Using the substitution \( u = \sin x \) and \( du = \cos x \, dx \), the integral becomes \( \int u^3 \, du \). This simplifies the integral significantly, turning it into a basic polynomial integral.

Integrate the polynomial

The integral \( \int u^3 \, du \) can be solved by using the power rule for integrals: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Applying this rule, we get: \( \frac{u^4}{4} + C \).

Substitute back to original variable

Replace \( u \) back with \( \sin x \) to express the result in terms of the original variable: \( \frac{\sin^4 x}{4} + C \). This is the final answer to the integral.

Key concepts

Substitution Method

The substitution method is a powerful technique in calculus used to simplify integrals by making a suitable substitution that transforms the original integral into a simpler form. This technique is especially useful when dealing with integrals containing composite functions. The key idea is to change the variable to simplify the expression.

Here's how the substitution method works in practice:

• **Identify a Suitable Substitution**: Look for a function and its derivative within the integral. In our example, we notice that the expression involves \( \sin^3 x \cos x \). By setting \( u = \sin x \), we match the derivative with the remaining part of the integral: \( du = \cos x \, dx \).
• **Rewrite the Integral**: Substitute \( \sin x \) with \( u \) and \( \cos x \, dx \) with \( du \). This converts the integral from a complicated trigonometric one to a simpler polynomial form, \( \int u^3 \, du \).
• **Evaluate the New Integral**: Solve the simpler integral using basic integration techniques, like the power rule.
• **Substitute Back**: After finding the antiderivative in terms of \( u \), revert back to the original variable to express the final answer in terms of \( x \).

The substitution method simplifies integration, making it approachable even when the integrand seems complex initially.

Trigonometric Integrals

Trigonometric integrals involve integrals of functions that include trigonometric functions such as sine, cosine, tangent, and others. These types of integrals often require specific techniques for successful evaluation. Understanding how to handle trigonometric functions is crucial when they appear in integrals.

When dealing with trigonometric integrals, consider the following:

**Characteristic Patterns**: Certain patterns often hint toward specific techniques. For example, the product of a power of sine and cosine, like \( \sin^n x \cos x \), can suggest potential substitution.

**Symmetry and Identities**: Using trigonometric identities, such as \( \sin^2 x + \cos^2 x = 1 \), can help reframe the integral into a more familiar form, potentially revealing easier paths to integration.

**Reduction Formulas**: In some cases, applying reduction formulas can repeatedly break the integral into simpler parts until it becomes solvable.

In our example, by recognizing the pattern and using substitution, we efficiently solve \( \int \sin^3 x \cos x \, dx \) through a straightforward transformation.

Integration by Parts

Integration by parts is another fundamental technique for tackling integrals, especially when they consist of products of functions that don't fit substitution easily. It's based on the product rule of differentiation and is formalized in the integration formula:\[\int u \, dv = uv - \int v \, du\]
Here’s how to apply integration by parts:

• **Select \( u \) and \( dv \)**: Choose \( u \) and \( dv \) from the original integral such that differentiation simplifies \( u \) and integration of \( dv \) is straightforward.
• **Differentiate and Integrate**: Compute \( du \) by differentiating \( u \) and find \( v \) by integrating \( dv \).
• **Apply the Formula**: Use the formula to transform the original integral into a potentially simpler form. Repeat the process if necessary until an easily solvable integral is reached.

While the method wasn’t necessary for our integral, it's an essential tool for more complex integrations, complementing the substitution method effectively in many problem-solving scenarios.