Question

Perform the indicated integrations. $$ \int\left(\sin ^{3} 2 t\right) \sqrt{\cos 2 t} d t $$

Short answer

The integral is \(-\frac{1}{3}(\cos 2t)^{3/2} + \frac{1}{7}(\cos 2t)^{7/2} + C\).

Step-by-step solution

Rewrite Using Trigonometric Identity

To solve the given integral \( \int (\sin^3 2t) \sqrt{\cos 2t} \, dt \), we start by expressing \( \sin^3 2t \) in a form that utilizes trigonometric identities. Recall that \( \sin^2 \theta = 1 - \cos^2 \theta \). Thus, \( \sin^3 2t \) can be rewritten as \( \sin 2t \cdot (1 - \cos^2 2t) \).

Split the Integral

The integral becomes \( \int (\sin 2t)(1 - \cos^2 2t) \sqrt{\cos 2t} \, dt \). Distribute the \( \sin 2t \), which results in two separate integrals: \( \int \sin 2t \sqrt{\cos 2t} \, dt - \int \sin 2t \cos^2 2t \sqrt{\cos 2t} \, dt \).

Simplify and Set Up Substitution

Let's perform substitution for both integrals. Let \( u = \cos 2t \). Thus, \( du = -2 \sin 2t \, dt \), or \(-\frac{1}{2} du = \sin 2t \, dt \). Substitute these into both integrals to ease computation.

Evaluate First Integral

For the first integral, \( \int \sin 2t \sqrt{\cos 2t} \, dt \) becomes \( -\frac{1}{2} \int u^{1/2} \, du \). Integrating \( u^{1/2} \) gives the result \( -\frac{1}{2} \cdot \frac{2}{3} u^{3/2} = -\frac{1}{3} u^{3/2} \). Convert back to \( t \) to get \( -\frac{1}{3} (\cos 2t)^{3/2} \).

Evaluate Second Integral

For the second integral, \( \int \sin 2t \cos^2 2t \sqrt{\cos 2t} \, dt \) becomes \( -\frac{1}{2} \int u^{5/2} \, du \). Integrating \( u^{5/2} \) gives \( -\frac{1}{2} \cdot \frac{2}{7} u^{7/2} = -\frac{1}{7} u^{7/2} \). Convert back to \( t \) to get \( -\frac{1}{7} (\cos 2t)^{7/2} \).

Combine Results and Simplify

Combine the results from the integrals: \(-\frac{1}{3} (\cos 2t)^{3/2} - (-\frac{1}{7} (\cos 2t)^{7/2})\). This simplifies to \(-\frac{1}{3} (\cos 2t)^{3/2} + \frac{1}{7} (\cos 2t)^{7/2} + C\), where \( C \) is the constant of integration.

Key concepts

Trigonometric Identities

Trigonometric identities are invaluable tools in calculus, especially when dealing with integration problems. They allow us to transform complex expressions into simpler forms. In our exercise, we encounter the integral \( \int (\sin^3 2t) \sqrt{\cos 2t} \, dt \). By applying the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), which states that the square of the sine function can be expressed in terms of the cosine function, we rewrite \( \sin^3 2t \) as \( \sin 2t \cdot (1 - \cos^2 2t) \). This change enables us to break the expression into parts that are more manageable for integration.
Using these identities not only simplifies the process but also breaks down the integral into parts that can be approached using different integration techniques. Remember:

• Trig identities help in simplifying complex problems.
• They provide alternative expressions for sine and cosine terms.

With trigonometric identities, you can transform an integral into a more approachable version by re-expressing elements within the integrand.

Substitution Method

The substitution method, also known as u-substitution, is a powerful technique in integration that simplifies the process of finding antiderivatives. In simple terms, it's about changing variables to make the integral look simpler.
In our problem, after applying trigonometric identities, we have integrals like \( \int \sin 2t \sqrt{\cos 2t} \, dt \). To proceed, we use substitution by letting \( u = \cos 2t \). Consequently, the differential is \( du = -2 \sin 2t \, dt \), or \(-\frac{1}{2} du = \sin 2t \, dt \). This replacement transforms the integral into an easier-to-handle form in terms of \( u \).
The substitution method offers:

• A straightforward way to replace complex expressions with simpler ones.
• A path to potentially easier integrals.

Converting back using the original variable ensures the final solution is in terms of the same variable found in the original integral.

Integral Simplification

Integral simplification is a fundamental aspect of solving integrals, aiming to make them easier to integrate. After applying trigonometric identities and substitution in the given problem, the integral breaks down into parts that can be further simplified.
Our integral \( \int (\sin^3 2t) \sqrt{\cos 2t} \, dt \) is transformed first through trigonometric identities and then substitution, leaving us with expressions like \( -\frac{1}{2} \int u^{1/2} \, du \) and \( -\frac{1}{2} \int u^{5/2} \, du \). Integration of these simpler power functions can then be carried out straightforwardly using the power rule of integration, which states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), provided \( n eq -1 \).
By simplifying each component separately, we can solve even complicated integrals by reducing them to fundamental operations and recombining them at the end:

• Simplification is about reducing the complexity of integrals at each step.
• It involves applying rules of integration effectively.

This organized approach allows clearer problem-solving pathways towards finding the antiderivative.