Problem 5 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-22\) $$ \int \sin ^{3} x d x $$

Step-by-Step Solution

Answer

\(-\cos x - \frac{\cos^3 x}{3} + C\)

Step 1: Identify the Problem Type

We need to evaluate \( \int \sin^3 x \ dx \). This is an example of an integral involving an odd power of sine, which suggests using a trigonometric identity for simplification.

Step 2: Use Trigonometric Identity

We can use the identity \( \sin^2 x = 1 - \cos^2 x \) to rewrite the integrand. Hence, \( \sin^3 x = \sin x \cdot \sin^2 x = \sin x \cdot (1 - \cos^2 x) \).

Step 3: Substitute the Identity

We rewrite the integral as: \[ \int \sin^3 x \ dx = \int \sin x (1 - \cos^2 x) \ dx = \int (\sin x - \sin x \cos^2 x) \ dx. \]

Step 4: Break Down into Simpler Integrals

This leads to two separate integrals: 1. \( \int \sin x \ dx \) 2. \( \int \sin x \cos^2 x \ dx \)

Step 5: Solve the First Integral

\( \int \sin x \ dx \) is a basic integral resulting in \(-\cos x + C_1\).

Step 6: Use Substitution for Second Integral

For \( \int \sin x \cos^2 x \ dx \), use substitution. Let \( u = \cos x \), then \( du = -\sin x \ dx \). The integral becomes: \[ -\int u^2 \ du \].

Step 7: Integrate with Respect to \( u \)

Calculate the integral: \[ -\int u^2 \ du = -\left( \frac{u^3}{3} \right) + C_2 = -\frac{\cos^3 x}{3} + C_2. \]

Step 8: Combine Results and Write Final Answer

Combine the results from previous steps: \[ -\cos x - \frac{\cos^3 x}{3} + C. \] Combining constants, the final answer is \(-\cos x - \frac{\cos^3 x}{3} + C\).

Key Concepts

Trigonometric Substitution

When encountering integrals involving trigonometric functions like \(\int \sin^3 x \ dx\), one effective technique is **trigonometric substitution**. This allows us to simplify the integral using known trigonometric identities. - **Identify the Function Type:** In examples with odd powers of sine or cosine, a strategic simplification can make the problem manageable.- **Utilize Trigonometric Identities:** For this exercise, we recognize that \( \sin^2 x = 1 - \cos^2 x \). This allows us to express \(\sin^3 x\) as \(\sin x \cdot (1 - \cos^2 x)\).By substituting, we break down the integral into simpler parts that can be individually solved, significantly reducing complexity.

Integral Calculus

**Integral Calculus** is the branch of calculus that deals with finding the accumulation of quantities, like areas under curves. In this context, we aim to evaluate the integral \(\int \sin^3 x \ dx\).- **Breaking Down the Integral:** After substitution, we split the original integral into two separate, simpler integrals: * \(\int \sin x \ dx\) * \(\int \sin x \cos^2 x \ dx\)- **Basic Integral Solutions:** The first integral, \(\int \sin x \ dx\), is a standard problem with a known solution of \(-\cos x + C_1\).- **More Complex Integrals:** The second requires further manipulation involving substitution, like setting \(u = \cos x\) to solve \(-\int u^2 \, du\).These steps transform a seemingly challenging problem into a series of manageable ones, each with a straightforward solution.

Trigonometric Identities

**Trigonometric Identities** play a crucial role in simplifying calculus problems involving trigonometric functions.- **Simplification of Integrals:** Using identities such as \(\sin^2 x = 1 - \cos^2 x\), enable the transformation of complex expressions into simpler forms.- **Strategy in Integration:** For the given problem, re-expressing \(\sin^3 x\) in terms of sine and cosine allows integrals to be broken down into solvable parts without initially overwhelming substitution or differentiation.Understanding these identities and their applications not only simplifies integrals but also builds a deeper intuition for calculus, making it an indispensable tool for solving complex problems.