Problem 11 - Calculus Early Transcendentals

Question

Integrals of \(\sin x\) or \(\cos x\) Evaluate the following integrals. $$\int \cos ^{3} x d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of \(\cos^3 x\) with respect to \(x\). Answer: $$\int \cos^3 x \, dx = \frac{1}{2} (\sin x +\sin x + \frac{1}{3} \sin (3x)) + C$$

Step 1: Rewrite the integrand using the Power-Reduction Formula

We begin by rewriting the integrand using the Power-Reduction Formula. The Power-Reduction Formula states that: $$\cos^2 x = \frac{1 + \cos2x}{2}.$$ Now, we have \(\cos^3 x\) as the integrand, we will rewrite it as follows: $$\int \cos^3 x \, dx = \int \cos x \cdot \cos^2 x \, dx.$$

Step 2: Apply the Power-Reduction Formula

We will now apply the Power-Reduction Formula to \(\cos^2 x\): $$\int \cos x \cdot \cos^2 x \, dx = \int \cos x \cdot \frac{1 + \cos2x}{2} \, dx = \frac{1}{2} \int \cos x(1+\cos 2x) \, dx$$

Step 3: Distribute and split the integral

Distribute and simplify the integrand, then split it into two separate integrals: $$\frac{1}{2} \int \cos x(1+\cos 2x) \, dx = \frac{1}{2}\left(\int \cos x \, dx+ \int \cos x\cos 2x \, dx\right)$$

Step 4: Integrate \(\cos x\)

The first integral is the integral of \(\cos x \, dx\) which is a standard integral: $$\int \cos x \, dx = \sin x + C_1$$

Step 5: Integrate \(\cos x \cos 2x\) using the Product-to-Sum Formula

To evaluate the second integral, we will use the Product-to-Sum Formula: $$\cos A \cos B = \frac{1}{2}(\cos(A - B) + \cos(A + B))$$ Apply this formula with \(A=x\) and \(B=2x\), we have: $$\cos x \cos 2x = \frac{1}{2}(\cos(-x) + \cos(3x))$$ Now, integrate: $$\int \cos x \cos 2x \, dx = \frac{1}{2} \int (\cos(-x) + \cos(3x)) \, dx = \frac{1}{2}(-\sin(-x) + \frac{1}{3}\sin(3x)) + C_2$$

Step 6: Add the results and simplify

Combine the results from Step 4 and Step 5, and multiply by the coefficient \(\frac{1}{2}\): $$\frac{1}{2} \left(\sin x - \sin (-x) + \frac{1}{3} \sin (3x) \right)+C = \frac{1}{2} (\sin x +\sin x + \frac{1}{3} \sin (3x)) + C$$

Step 7: Final Result

The final result is: $$\int \cos^3 x \, dx = \frac{1}{2} (\sin x +\sin x + \frac{1}{3} \sin (3x)) + C$$

Key Concepts

Power-Reduction Formula

When integrating trigonometric functions, especially when they involve powers of functions like \( \sin x \) or \( \cos x \), it's helpful to use reduction techniques to simplify the integral. The Power-Reduction Formula is a handy tool in these scenarios. This formula allows one to express higher powers of trigonometric functions in terms of their lower powers, by leveraging identities that relate squared functions to linear trigonometric expressions.

For instance, the Power-Reduction Formula states that:

• \( \cos^2 x = \frac{1 + \cos 2x}{2} \)
• \( \sin^2 x = \frac{1 - \cos 2x}{2} \)

Integrating \( \cos^3 x \) requires us to split it as \( \cos x \cdot \cos^2 x \). Applying the Power-Reduction Formula to \( \cos^2 x \) simplifies the problem into a linear combination of integrals, thus making the calculus much more manageable.

By substituting \( \cos^2 x \) with its equivalent from the Power-Reduction Formula, this approach simplifies the original problem into a form that can be further dissected and integrated using basic calculus techniques.

Product-to-Sum Formula

Trigonometric integrals often involve products of sine and cosine functions which can be complex to handle. The Product-to-Sum Formula is crucial in these cases as it helps convert the product of trigonometric functions into a sum or difference of two easier expressions. This simplification is key to solving integrals that otherwise seem daunting.

The Product-to-Sum Formula is defined as:

• \( \cos A \cos B = \frac{1}{2}(\cos(A - B) + \cos(A + B)) \)

In the exercise, to evaluate \( \int \cos x \cos 2x \, dx \), this formula plays a pivotal role. By translating the product into a sum, the integral splits into manageable parts: \( \frac{1}{2} \int (\cos(-x) + \cos(3x)) \, dx \). Each of these parts can then be integrated using standard trigonometric integration techniques.
This formula effectively reduces complexity and makes handling integrals involving products of cosines much more feasible, providing an elegant solution pathway where direct integration isn't straightforward.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions that are composed of trigonometric functions such as sine, cosine, tangent, etc. These can be intricate because trigonometric functions often interact with each other in complicated ways through multiplication or through their powers.

For integrals involving products or powers, techniques like rewriting using identities (such as Power-Reduction or Product-to-Sum) and strategic substitution are crucial. In the case of \( \cos^3 x \), initially addressing \( \cos^2 x \) separately using identities is an effective way to break down the complexity.

Understanding the standard integrals of sine and cosine like \( \int \cos x \, dx = \sin x + C \) or \( \int \sin x \, dx = -\cos x + C \) forms the foundation. By incorporating these with product identities, we can efficiently determine solutions for more convoluted expressions, ultimately arriving at the final integral result.
Being systematic in applying these methods unlocks the solution to integrate any standard trigonometric function, helping students become adept at maneuvering through otherwise challenging calculus problems.