Question

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \cos ^{5} x d x\)

Short answer

Use substitution and power reduction formulas to integrate cosine functions with odd powers.

Step-by-step solution

Apply the Power Reduction Formula

To integrate powers of cosine, we can use the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \). However, for odd powers like \( \cos^5 x \), it is helpful to split off a single cosine: \( \cos^5 x = \cos^4 x \cos x = (\cos^2 x)^2 \cos x \).

Simplify the Expression

Using the identity for \( \cos^2 x \) again, rewrite \( (\cos^2 x)^2 \) as \( (\frac{1+\cos(2x)}{2})^2 \). Thus, the expression becomes \[ \left( \frac{1+\cos(2x)}{2} \right)^2 \cos x \].

Substitute and Simplify Further

Now use substitution \( u = \sin x \), which implies \( du = \cos x \, dx \). Substitute into the integral: \[ \int \left( \frac{1+\cos(2\arccos(u))}{2} \right)^2 du \]. Simplify further if needed.

Expand and Integrate

Expand the squared expression: \[ \int \left( \frac{1}{4} + \frac{1}{2} \cos(2\arccos(u)) + \frac{1}{4}\cos^2(2\arccos(u)) \right) du \]. Use the half-angle identities or look up the integral in a table, if necessary.

Evaluate the Integral

After expansion, each term can now be integrated separately. Let's compute the integrals for each part. For example, \( \int \frac{1}{4} \, du = \frac{1}{4}u \). Follow similar steps for the other terms.

Substitute Back

After evaluating the integral in terms of \( u \), substitute back \( u = \sin x \).For instance, the integral \( \int \frac{1}{4} du \) gives \( \frac{1}{4} \sin x \), and similar computations are needed for remaining terms.

Add Constants of Integration and Simplify

Combine all parts of the antiderivative and add a constant of integration, \( C \). Simplify to consolidate integration constants.

Key concepts

Power Reduction Formula

When dealing with trigonometric integrals, particularly those involving powers of sine or cosine, the **Power Reduction Formula** becomes invaluable. This formula allows you to rewrite trigonometric functions that are raised to even or odd powers into forms that are easier to integrate. For cosine, the reduction formula is: \[ \cos^2 x = \frac{1 + \cos(2x)}{2} \]In the exercise \(\int \cos^5 x \, dx\), the odd power leads us to separate a \(\cos x\) term. This isolates the single cosine and helps in breaking down the remaining even power through the power reduction formula. By expressing higher even powers using the reduction formula, the integrals become more manageable. For instance, rewriting \(\cos^4 x\) as \((\cos^2 x)^2\) sets the stage for application of the formula, turning it into \(\left(\frac{1 + \cos(2x)}{2}\right)^2\).This stage simplifies the function enough to be integrated by other methods, such as substitution or directly, depending on the complexity of the resulting expression.

Substitution Method

The **Substitution Method** is a crucial technique in calculus for integrating complex expressions. It involves changing variables to simplify the integration process. This is akin to finding patterns that mirror the chain rule, used in differentiation.In the exercise involving \(\int \cos^5 x \, dx\), after simplifying using the power reduction formula, substitution is key for proceeding further. By letting \(u = \sin x\), we take advantage of the relationship \(du = \cos x \, dx\). This substitution works well because it directly simplifies the integral by leveraging the differential \(du\) to replace \(\cos x \, dx\).This transforms the integral into a function of \(u\), making it much simpler to handle. After solving the integral in terms of \(u\), it’s important to revert back to the original variable \(x\) by substituting \(u = \sin x\) back in. This restores the problem to its original context while providing the solution needed.

Half-angle identities

**Half-angle identities** play an essential role in simplifying trigonometric expressions for integration. These identities allow further reduction and transformation of trigonometric functions, converting them into powers that are more manageable within integration.While working on an integral such as \(\int \cos^5 x \, dx\), you may encounter expressions requiring further transformation. For instance, expressions like \(\cos(2x)\) or higher powers might crop up during calculations. The half-angle identities are particularly useful here:

• \( \sin^2 x = \frac{1 - \cos(2x)}{2} \)
• \( \cos^2 x = \frac{1 + \cos(2x)}{2} \)

These identities are crucial for handling squared trigonometric functions, either simplifying integrals directly or breaking down complex expressions into integrable parts. By applying half-angle transformations, the integration of trigonometric functions becomes significantly easier, ensuring clearer and more straightforward solutions.

Integration of powers of trigonometric functions

The **integration of powers of trigonometric functions** requires specialized approaches due to their repetitive and oscillatory nature. High expressions of powers like \(\cos^5 x\), present challenges that can be cleverly simplified using techniques mentioned earlier.For odd powers, like in \(\cos^5 x\), rewriting the function with one separated trigonometric term, (i.e., \(\cos x \)), allows the remaining even power to be reduced using identities such as the power reduction formula. For higher even powers, derivatives and transformations using reduction or half-angle identities are typically put to use.The goal is always to reduce complex trigonometric powers into more elementary forms, like linear or even simple quadratic polynomials in other trigonometric identities. These simplified forms are either directly integrable or manageable through substitution. In all scenarios, mastering these techniques opens pathways to efficiently address integrations of these diverse and oscillatory functions.