Question

Evaluate the indefinite integral $$ \int \sin ^{3}\left[\left(x^{2}+1\right)^{4}\right] \cos \left[\left(x^{2}+1\right)^{4}\right]\left(x^{2}+1\right)^{3} x d x $$ Hint: Let \(u=\sin \left(x^{2}+1\right)^{4}\).

Short answer

\( \frac{\sin^4\left(\left(x^2+1\right)^4\right)}{32} + C \)

Step-by-step solution

Identify the Substitution

The hint suggests substituting \( u = \sin \left(\left(x^2 + 1\right)^4\right) \). This will simplify the integral by transforming products of trigonometric functions into a simpler form.

Differentiate to Find \( du \)

Differentiate \( u \) with respect to \( x \). First use the chain rule: \( \frac{du}{dx} = \cos\left(\left(x^2+1\right)^4\right) \cdot 4\left(x^2+1\right)^3 \cdot 2x \). Simplify to find \( du = \cos\left(\left(x^2+1\right)^4\right) 8x\left(x^2+1\right)^3 dx \).

Simplify the Integral

Notice the integral contains \( \cos\left(\left(x^2+1\right)^4\right)\left(x^2+1\right)^3 x \), which partially matches \( du \). Rewrite the integral as \( \int u^3 \cdot \frac{1}{8} du \).

Perform the Integration

Since the integral is now \( \int u^3 \cdot \frac{1}{8} du \), factor out the constant to get \( \frac{1}{8}\int u^3 \, du \). Integrate to get \( \frac{1}{8} \cdot \frac{u^4}{4} + C = \frac{u^4}{32} + C \), where \( C \) is the constant of integration.

Back-Substitution

Substitute back \( u = \sin\left(\left(x^2 + 1\right)^4\right) \) into the antiderivative: \( \frac{\sin^4\left(\left(x^2+1\right)^4\right)}{32} + C \).

Key concepts

Integration by Substitution

Integration by substitution is a method used to simplify the process of finding integrals. This technique is particularly useful when dealing with complex functions such as nested trigonometric functions.
The idea behind substitution is to change variables in a way that makes the integral easier to solve. Typically, you will choose a substitution that simplifies the integrand, transforming a difficult integral into one that can be evaluated more straightforwardly.
In the given exercise, we used the substitution \( u = \sin \left( (x^2 + 1)^4 \right) \) to streamline the integration process.

• Selecting the right substitution is crucial. It should simplify the expression inside the integral significantly.
• After choosing \( u \), we differentiate \( u \) to express \( du \) in terms of \( x \) and \( dx \), which allows us to rewrite the original integral in a simpler form.

By substituting, complex expressions involving \( x \) are replaced with simpler expressions involving \( u \), making the integration process more manageable.

Chain Rule

The chain rule is a fundamental tool in calculus that helps us differentiate composite functions.
It is essential to understand how to apply the chain rule, particularly when working with functions nested inside one another, like in the exercise where \( u = \sin((x^2 + 1)^4) \).

• The chain rule states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outer function \( f \) by the derivative of the inner function \( g \).
• In practice, this means \( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \).

For our exercise, applying the chain rule follows these steps:

• Differentiating \( \sin((x^2 + 1)^4) \) requires considering \((x^2 + 1)^4\) as the inner function.
• We first find the derivative of \(\sin(u)\), which is \(\cos(u)\), and then multiply by the result of differentiating \((x^2 + 1)^4\).
• Differentiating \((x^2 + 1)^4\) involves using the chain rule again: \( 4(x^2 + 1)^3 \cdot 2x \).

Overall, the chain rule enables us to break down and manage the complexity of differentiating nested functions.

Trigonometric Functions

Trigonometric functions like \( \sin \) and \( \cos \) are often involved in calculus problems, especially in integration and differentiation.
Understanding how these functions and their derivatives behave is key to tackling calculus efficiently.

• The derivative of the sine function \( \sin(x) \) is \( \cos(x) \).
• The derivative of the cosine function \( \cos(x) \) is \(-\sin(x) \).

In integrating functions involving trigonometric expressions, like our exercise, knowing these derivatives is essential, particularly when applying methods like the chain rule.
When you encounter products or compositions like \( \sin \) and \( \cos \), the aim is to simplify the expression using techniques such as integration by substitution.
Trigonometric identities and properties often aid in simplifying complicated integrals. Understanding and being able to manipulate these identities is key in breaking down difficult problems.

Antiderivatives

Finding the antiderivative, or the indefinite integral, is essentially the reverse process of differentiation.
The goal is to find a function whose derivative is the given function, adding an arbitrary constant \( C \) to account for all possible vertical shifts of the graph of the antiderivative.

• In our problem, after simplifying the integral with substitution, integrating \( u^3 \) yields the antiderivative \( \frac{u^4}{4} \).
• The constant factor \( \frac{1}{8} \) is carried along and combined to give the result: \( \frac{u^4}{32} + C \).

Antiderivatives are fundamental in calculus because they allow us to calculate areas under curves and solve differential equations.
After finding the antiderivative using substitution, don't forget the back-substitution that replaces \( u \) with the original expression. This step transforms the result back into terms of \( x \), providing the final answer in the original context of the problem.