Question

Perform the indicated integrations. $$ \int_{0}^{\pi / 2} \sin ^{6} \theta d \theta $$

Short answer

The result of the integral is \( \frac{\pi}{16} \).

Step-by-step solution

Use a Trigonometric Identity

To integrate \( \sin^6 \theta \), we can use the identity for powers of sine: \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \). Substitute this identity into the integral.

Simplify the Integral

Using the identity, rewrite \( \sin^6 \theta \) as \( (\sin^2 \theta)^3 = \left( \frac{1 - \cos(2\theta)}{2} \right)^3 \). This expands to \( \frac{1}{8}(1 - 3\cos(2\theta) + 3\cos^2(2\theta) - \cos^3(2\theta)) \).

Expand Using Further Identities

Use \( \cos^2(2\theta) = \frac{1 + \cos(4\theta)}{2} \) and for \( \cos^3(2\theta) \) use \( \cos^3(2\theta) = \cos(2\theta) \times \cos^2(2\theta) \). Substitute these into the integral.

Break Down the Integral

Split the integral into separate terms: \[\int_0^{\pi/2} \frac{1}{8} (1) \,d\theta - \int_0^{\pi/2} \frac{3}{8} \cos(2\theta) \,d\theta + \int_0^{\pi/2} \frac{3}{16} \cos(4\theta) \,d\theta - \int_0^{\pi/2} \frac{1}{8} \cos^3(2\theta) \,d\theta \].

Evaluate Each Integral

Evaluate each integral separately:1. \( \int_0^{\pi/2} \frac{1}{8} \,d\theta \) gives \( \frac{1}{8} \times \frac{\pi}{2} = \frac{\pi}{16} \).2. \( \int_0^{\pi/2} \frac{3}{8} \cos(2\theta) \,d\theta = 0 \) since the integral of cosine over a full period is zero.3. \( \int_0^{\pi/2} \frac{3}{16} \cos(4\theta) \,d\theta = 0 \) for the same reason of periodicity.4. For \( \int_0^{\pi/2} \frac{1}{8} \cos^3(2\theta) \,d\theta \), use a reduction formula or symmetry argument to determine it is zero.

Combine Results

Combine the results from the previous step: \( \frac{\pi}{16} + 0 + 0 - 0 = \frac{\pi}{16} \).

Key concepts

Trigonometric Identities

Trigonometric identities are crucial mathematical tools that simplify complex trigonometric expressions. In the context of integrating powers of sine, we often use the identity \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \). This particular identity is useful because it expresses the square of sine in terms of cosine, which can be further manipulated using standard integrations or other identities. When dealing with powers greater than two, such as \( \sin^6 \theta \), we can repeatedly apply basic identities to reduce the power. For instance, \( \sin^6 \theta = (\sin^2 \theta)^3 \) is handled by first reducing \( \sin^2 \theta \) using the identity, substituting back, and then expanding to facilitate easier integration. Utilizing trigonometric identities in calculus not only makes the integrals more manageable but also highlights the inter-relationship between different trigonometric functions, particularly sine and cosine, which are often transformed using the double angle formulas.

Integration Techniques

Integration techniques refer to various methods and strategies applied to simplify and evaluate integrals. In our problem involving \( \int_{0}^{\pi / 2} \sin^6 \theta \, d\theta \), we use the strategy of substitution. Starting with the trigonometric identity, we substitute \( \sin^2 \theta \) to reduce the expression to functions of cosine that are solvable.After substitution, expansion is necessary. The integral results in multiple terms that are simpler to calculate individually. Addressing each as a separate integral, we can evaluate them using fundamental integration rules. For example, recognizing certain integrals like \( \int \cos(k\theta) \, d\theta \) yield zero over complete periods due to the cosine function's symmetry and periodicity. These steps showcase how breaking down a complex integral into smaller, more tractable parts can simplify solving even higher-power trigonometric integrals. Pairing substitution with periodic properties is a standard integration technique that often makes the difference between a solvable and unsolvable problem.

Reduction Formulas

Reduction formulas are specialized equations used to reduce the degree of integrals of powers, thereby simplifying complex expressions into more manageable forms. Specifically, when faced with an integral like \( \int \cos^3(2\theta) \, d\theta \), one might utilize recurrence or reduction formulas, sometimes even depending on related identities like the double angle formulas.In this context, a "reduction" effectively implies the use of a formula to express the integral of a higher power function in terms of the integral of a lower power function. For cosine or sine functions raised to periodic powers, it's common these can simplify to zero over a symmetric interval—for example, due to symmetry from \( 0 \) to \( \pi/2 \). This means for integrals like \( \int \cos^3(2\theta) \, d\theta \), either leveraging a known reduction formula or using symmetry arguments considerably simplifies evaluation.Reduction formulas demand familiarity with identities and properties of trigonometric functions, often turning what seems an insurmountable integration challenge into a straightforward calculation. These are particularly helpful when other integration techniques fall short or become cumbersome.