Problem 33 - Calculus

Question

Use the method of substitution to find each of the following indefinite integrals. $$ \int x^{2} \sin \left(x^{3}+5\right) \cos ^{9}\left(x^{3}+5\right) d x $$

Step-by-Step Solution

Answer

\(-\frac{1}{30} \cos^{10}(x^3 + 5) + C\)

Step 1: Identify the Substitution

First, identify the part of the integral that can be substituted with a single variable. Here, notice that the expression inside the sine and cosine functions, \(x^3 + 5\), is a good candidate for substitution. Let's set \(u = x^3 + 5\).

Step 2: Differentiate the Substitution

Differentiate the substitution equation \(u = x^3 + 5\) with respect to \(x\) to find \(du\). This gives \(du = 3x^2 \, dx\).

Step 3: Solve for dx

From \(du = 3x^2 \, dx\), solve for \(dx\) to use in the integral. Thus, \(dx = \frac{du}{3x^2}\).

Step 4: Rewrite the Integral in Terms of u

Replace \(x^2\) and \(dx\) in the integral with terms involving \(u\). Since \(x^2 \cdot dx = \frac{du}{3}\), we can write:\[\int x^2 \sin(u) \cos^9(u) \frac{du}{3}\].

Step 5: Simplify the Integral

Simplify the integral by factoring out constant coefficients. This results in:\[\frac{1}{3} \int \sin(u) \cos^9(u) \, du\].

Step 6: Solve the New Integral

Now, integrate \(\sin(u) \cos^9(u)\) with respect to \(u\). Using the power reduction formula or any suitable method for odd powers of sine and cosine, solve the integral. Here, let's assume the formula:\[\int \sin(u) \cos^n(u) du = -\frac{1}{n+1} \cos^{n+1}(u) \].Applying it gives:\[-\frac{1}{10} \cos^{10}(u)\].

Step 7: Re-substitute x into the Result

Replace \(u\) back with \(x^3 + 5\) to express the result in terms of \(x\). Therefore, the integrated result is:\[ -\frac{1}{30} \cos^{10}(x^3 + 5) + C \], where \(C\) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify the process of integrating complex functions. It involves replacing a part of an integral with a new variable to make the integration easier. In this exercise, we have the integral \( \int x^{2} \sin(x^{3}+5) \cos^{9}(x^{3}+5) \, dx \).
Firstly, we identify a part of the integral to substitute. We choose the expression within the trigonometric functions, \( x^3 + 5 \), because it can be simplified to a variable \( u \). In this way, our new integral becomes a function of \( u \) rather than \( x \).
After substituting, we differentiate \( u = x^3 + 5 \) to find \( du = 3x^2 \, dx \). This relationship allows us to replace \( dx \) in terms of \( du \) and \( x \). The key to successful substitution is ensuring that all terms originally in terms of \( x \) are transformed into terms of \( u \).
Substitution streamlines the integral into a simpler form, making it easier to evaluate. This method relies on the ability to choose an expression inside the integral that will simplify considerably when differentiated, allowing for an ease in calculations.

Integration Techniques

When faced with integrating complex expressions, we can use various integration techniques to solve them. The substitution method, as utilized in this example, is one of these effective strategies.
To begin using this technique, it's always important to identify a part of the integral which when substituted, simplifies the integral. Here, \( u = x^3 + 5 \) was used because it simplifies both \( \, \sin(x^{3}+5) \) and \( \, \cos^{9}(x^{3}+5) \) into more manageable terms when integrated.
After substitution, we replace \( dx \) using our derived \( du = 3x^2 dx \), allowing us to rewrite the initial integral with respect to \( u \). This often involves solving for the differential \( dx \) in terms of \( du \) to integrate smoothly.
Simplifying the integral by factoring out constants also eases the calculation. For instance, the given problem involves pulling out a factor of \( \frac{1}{3} \) since \( dx \) entails a division by \( 3 \). Techniques such as these allow us to handle otherwise difficult integrals with more straightforward calculus operations.

Trigonometric Functions

Trigonometric functions like sine and cosine play a significant role in various integrals, especially when they appear in conjunction with exponentials or powers. In this particular integral, we have \( \sin(x^3 + 5) \) and \( \cos^9(x^3 + 5) \) which arise from substituting \( u = x^3 + 5 \).
Recognizing patterns such as odd and even powers in trigonometric functions is crucial. Here, \( \cos^9(u) \) demonstrates a typical situation where an odd power allows us to apply certain integration techniques efficiently. The odd power indicates that a power-reduction formula or a simple power rule can be applied after substitution reduces the complex expressions to basic trigonometric forms in \( u \).
In our solution, we use a known result for the integral of \( \sin(u) \cos^n(u) \), which facilitates the integration of trigonometric expressions. By understanding these properties and identities of trigonometric functions, complex integrals involving these can become more approachable and solvable.