Problem 280 - Calculus Volume 1

Question

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t^{2} \cos ^{2}\left(t^{3}\right) \sin \left(t^{3}\right) d t $$

Step-by-Step Solution

Answer

\(-\frac{1}{9} \cos^3(t^3) + C\)

Step 1: Choose the substitution variable

Look at the integral and identify a substitution that simplifies the integrand. Here, notice the inner function of the trigonometric functions is \(t^3\). So, we choose \(u = t^3\).

Step 2: Compute the differential

Differentiate the variable \(u = t^3\) to express \(dt\) in terms of \(du\). We have \(du = 3t^2 dt\) or equivalently, \(dt = \frac{du}{3t^2}\).

Step 3: Substitute and simplify the integral

Substitute \(u = t^3\) and \(dt = \frac{du}{3t^2}\) into the integral. The integrand becomes \(t^2 \cos^2(u) \sin(u) \cdot \frac{du}{3t^2}\). The \(t^2\) terms cancel out, simplifying to \(\frac{1}{3} \int \cos^2(u) \sin(u) \, du\).

Step 4: Integrate using substitution

Recognize that \(\int \cos^2(u) \sin(u) \, du\) can be computed using the substitution \(v = \cos(u)\), hence \(dv = -\sin(u) \, du\). Rewrite the integral as \(-\int v^2 \, dv\), which integrates to \(-\frac{v^3}{3} + C = -\frac{\cos^3(u)}{3} + C\).

Step 5: Undo the substitution

Back-substitute \(u = t^3\) into the result from Step 4 to revert to the variable \(t\). The indefinite integral is \(-\frac{1}{9} \cos^3(t^3) + C\).

Key Concepts

Substitution Method

The substitution method is a powerful tool used in calculus to simplify complex integrals. The main idea is to replace a complex part of the integrand with a single variable. This makes the integral easier to solve. This method is especially useful for integrals involving composite functions. Here, we change the variable to simplify the process.For example, in the integral \( \int t^{2} \cos ^{2}\left(t^{3}\right) \sin \left(t^{3}\right) dt \), the substitution is \( u = t^3 \). This choice simplifies the expression inside the trigonometric functions.

• Identify a part of the integrand as a function \( u \).
• Differentiate to find \( du \).
• Replace all instances of the original variable and \( dt \) with \( u \) and \( du \).

By simplifying the integrand, the substitution method allows for easier computation. This approach, however, requires reversing the substitution once integrated.

Change of Variables

The change of variables technique, closely related to the substitution method, changes the integrand into terms of a new variable. This method can simplify solving the integral by transforming it into a more straightforward format. In our example, after recognizing the inner function \( t^3 \), we set \( u = t^3 \). We then find that \( du = 3t^2 dt \), or equivalently, solve for \( dt \) as \( dt = \frac{du}{3t^2} \). This modification allows us to cancel out complex parts of the integrand.

• This requires careful differentiation and substitution.
• All original variables should be in terms of the new variable \( u \).

Using a change of variables can make complicated functions easier to handle. After integrating, it's important to substitute back to the original variable to obtain the final answer.

Integration Techniques

Integration techniques are various strategies used to solve integrals. They include substitution, integration by parts, partial fraction decomposition, and more. The choice of technique depends on the form of the integrand. Our main focus here is substitution, which simplifes complex expressions.After substituting \( u = t^3 \) into the integral, we simplify it to \( \frac{1}{3} \int \cos^2(u) \sin(u) \, du \). Recognizing that this integral is suitable for further substitution, we choose the substitution \( v = \cos(u) \). This changes the integral into a standard polynomial form, \( -\int v^2 \, dv \).

• Effective integration requires recognizing patterns and suitable substitutions.
• Simplified integration often involves computing polynomial-like integrals.

By leveraging these techniques, previously daunting integrals become manageable, often resulting in simpler polynomial expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. These identities are useful in simplifying trigonometric integrals by transforming them into more familiar forms. They play a key role in the integration process.In our integral \( \int t^{2} \cos ^{2}\left(t^{3}\right) \sin \left(t^{3}\right) dt \), trigonometric identities don't directly simplify things at first glance. However, through substitution, we bypass direct manipulation using identities.

• Trigonometric identities, such as \( \sin^2(x) + \cos^2(x) = 1 \), help in rearranging terms.
• In certain scenarios, they offer alternative paths to integrate functions.

Utilizing these identities effectively can turn a complex trigonometric integral into a manageable one, but in this case, substitution provided the key simplifying step.