Question

Perform the indicated integrations. $$ \int \cos ^{6} \theta \sin ^{2} \theta d \theta $$

Short answer

The integral simplifies to \( -\frac{1}{8} \cos \theta (56 \theta - 48 \sin \theta + 18 \sin 3\theta - \sin 5\theta) + C \).

Step-by-step solution

Recognize the Integration Technique

To solve the integral \( \int \cos^6 \theta \sin^2 \theta \, d \theta \), we'll use integration techniques suited for trigonometric expressions. One effective way is to use trigonometric identities to simplify the expression.

Apply Trigonometric Identity

The expression \( \cos^6 \theta \) can be rewritten using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \). Thus, \( \cos^6 \theta = (\cos^2 \theta)^3 = (1 - \sin^2 \theta)^3 \). This simplifies our integral to:\[ \int (1 - \sin^2 \theta)^3 \sin^2 \theta \, d\theta \]

Expand the Expression

Expand \( (1 - \sin^2 \theta)^3 \) using the binomial theorem to get:\[ 1 - 3 \sin^2 \theta + 3 \sin^4 \theta - \sin^6 \theta \]Substitute this back into the integral:\[ \int (1 - 3 \sin^2 \theta + 3 \sin^4 \theta - \sin^6 \theta) \sin^2 \theta \, d \theta \]

Distribute \( \sin^2 \theta \) Across the Expression

Distribute \( \sin^2 \theta \) across the expanded expression:\[ \int (\sin^2 \theta - 3 \sin^4 \theta + 3 \sin^6 \theta - \sin^8 \theta) \, d \theta \]Now, split this into separate integrals:\[ \int \sin^2 \theta \, d \theta - 3 \int \sin^4 \theta \, d \theta + 3 \int \sin^6 \theta \, d \theta - \int \sin^8 \theta \, d \theta \]

Solve Each Integral Separately

Solve each of these integrals using standard integration techniques:1. \( \int \sin^2 \theta \, d \theta \)2. \( \int \sin^4 \theta \, d \theta \)3. \( \int \sin^6 \theta \, d \theta \)4. \( \int \sin^8 \theta \, d \theta \)These often involve using double angle identities like \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \) and reduction formulas.

Combine Results

Combine the integrated results from the previous step. This part often involves algebraic simplification after finding the indefinite integrals. Integrals of higher powers of sine can typically be reduced using known integral formulas:\[ \int \sin^n \theta \, d\theta = \frac{-1}{n} \sin^{n-1} \theta \cos \theta + \frac{n-1}{n} \int \sin^{n-2} \theta \, d\theta \]After thorough calculation, combine these to obtain the final expression.

Final Expression and Constant

Make sure to include the constant of integration \( C \) at the end of the integration process. The combined integral gives us the antiderivative:\[ -\frac{1}{8} \cos \theta (56 \theta - 48 \sin \theta + 18 \sin 3\theta - \sin 5\theta) + C \]

Key concepts

Integration Techniques

Integration techniques are powerful tools to solve a variety of integral problems. One critical technique for trigonometric integrals, like the one given here, involves simplifying the integrand.
In problems involving powers of sine and cosine, a wise strategy is to reduce the complexity using identities.
Integration by substitution or trigonometric identities can simplify the integral into a more manageable form. In our exercise, we began by recognizing that the expression could be expanded using identities before integrating. This preparation makes the actual integration more straightforward.
This exercise demonstrates how to apply trigonometric identities and other methods to integrate complex trigonometric functions effectively.

Trigonometric Identities

Trigonometric identities are fundamental in simplifying integrals involving trigonometric functions. They offer a way to transform expressions into forms that are easier to integrate.
In our problem, the identity \( \cos^2 \theta = 1 - \sin^2 \theta \) was pivotal. By rewriting \( \cos^6 \theta \) as \, \( (\cos^2 \theta)^3 \), which is \( (1 - \sin^2 \theta)^3 \), we simplified the integral greatly.
The strategic use of trigonometric identities not only changes the form of the integral but also opens paths to employ other mathematical tools like the binomial theorem for further expansion.
Remember these identities are part of your mathematical toolkit, ready to be used for simplification and problem-solving in integration tasks.

Binomial Theorem

The Binomial Theorem provides an efficient method for expanding powers of binomials. It was a crucial step in our exercise when handling the expression \( (1-\sin^2\theta)^3 \).
The theorem states that for any positive integer \( n \), the expansion of \( (x + y)^n \) is given by \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]Where \( \binom{n}{k} \) denotes the binomial coefficient.
In this exercise, \( x \) was 1 and \( y \) was \(-\sin^2\theta \), which was expanded and led to a more manageable integration.
By applying the binomial expansion, you can convert complex expressions into a sum of simpler terms that are easier to integrate.

Double Angle Identities

Double angle identities allow us to rewrite trigonometric functions of doubled angles in terms of simpler functions. These identities simplify integration processes, especially when dealing with trigonometric functions raised to powers.
One prominent identity used in integration of powers is \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \), which reduces power terms. Another key identity is \( \cos(2\theta) = 2\cos^2\theta - 1 \).
In the exercise, these identities can simplify or break down powers of sine and cosine into integrable parts. By converting power terms using double angle identities, we turned the complex trigonometric integrals into simpler terms, which could be then handled with basic integration techniques.