Problem 10 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Evaluate the given integral: \(\int \sin^{3} x dx\) Answer: \(\int \sin^{3} x dx = -\frac{1}{2}\sin x\cos(2x) - \frac{2}{3}\cos^3(x) + C\)

Step 1: Rewrite the integral expression

First, rewrite the given integral as: $$\int \sin^{3} x dx = \int \sin x\cdot \sin^2 x dx$$ And using the double-angle formula, we get \(\sin^2 x = \frac{1-\cos(2x)}{2}\). Therefore, the integral becomes: $$\int \sin x\cdot \frac{1-\cos(2x)}{2} dx$$

Step 2: Apply integration to \(\sin x\)

Now, let's define two variables, u and v: u = \(\cos(2x)\) and v = sin(x) We need to find the integral of sin(x), which is: $$\int v du$$ To find du, differentiate u: $$du = -2\sin(2x)dx$$ And dv is simply: $$dv = \cos(x)dx$$

Step 3: Integration by Parts

Now using integration by parts formula: $$\int u dv = uv - \int v du$$ Applying this formula to our integral, we get: $$-\frac{1}{2}\int \sin x (1 - \cos(2x)) dx = \frac{1}{2}\left[-\sin x\cos(2x) + 2\int \sin(2x)\sin x dx\right]$$

Step 4: Simplify the expression

Let's simplify the expression: $$-\frac{1}{2}\int \sin x (1 - \cos(2x)) dx = -\frac{1}{2}\sin x\cos(2x) + \int (\sin x\sin(2x)) dx$$

Step 5: Solve for the remaining integral using double angle formulas

Recall the following double-angle formula for sine: $$\sin(2x) = 2\sin(x)\cos(x)$$ Now, substitute the formula into the integral: $$\int (\sin x\sin(2x))dx = \int (2\sin^2(x)\cos(x))dx$$ Make a substitution by letting \(t = \cos(x)\), then \(-dt = \sin(x)dx\): $$\int (2\sin^2(x)\cos(x))dx = -2\int t^2 dt$$ Now we can easily integrate \(t^2\): $$-2\int t^2 dt = -\frac{2}{3}t^3 + C$$ Substitute the value of \(t\) back: $$-\frac{2}{3}t^3 + C = -\frac{2}{3}\cos^3(x) + C$$

Step 6: Combine the results

Now we can combine both parts of our solution: $$-\frac{1}{2}\int \sin x (1 - \cos(2x)) dx = -\frac{1}{2}\sin x\cos(2x) + \int (\sin x\sin(2x)) dx$$ So, we have the final result: $$\int \sin^{3} x dx = -\frac{1}{2}\sin x\cos(2x) - \frac{2}{3}\cos^3(x) + C$$

Key Concepts

Trigonometric Integrals

Trigonometric integrals are a special category of integrals that involve trigonometric functions like sine, cosine, and tangent. These problems typically require manipulating the expression using trigonometric identities to simplify the integration process.

When faced with integrals such as \( \int \sin^3 x \, dx \), you can break down the problem by expressing it as a product of simpler sine and cosine functions. This approach utilizes trigonometric identities to manage powers or combined trigonometric expressions efficiently.

To tackle such integrals, you often end up rewriting terms using standard identities. For example:

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

These identities are essential because they help simplify the integrals, making them easier to evaluate.

Integration by Parts

Integration by parts is a powerful technique derived from the product rule of differentiation. It’s particularly useful when you have an integrand that is a product of functions. The formula is:

\[ \int u \, dv = uv - \int v \, du \]

Choosing the right \( u \) and \( dv \) in your integration problem can make a big difference in simplicity. In the provided exercise, it was key to separate terms correctly. For instance, consider:

• \( u = \cos(2x) \)
• \( dv = \sin(x) \, dx \)

This choice leads to a manageable integration process, but you need to carefully differentiate and integrate to ensure the method works correctly. Upon solving the integral with these variables, the parts neatly come together to give you a final answer.

Double-Angle Formulas

Double-angle formulas are specific trigonometric identities that connect expressions involving double angles \( 2x \) with simpler single angle trigonometric expressions, like \( x \). These are particularly useful in integration because they can transform square terms into linear terms, making integration more feasible.

For example:

• \( \sin(2x) = 2\sin(x)\cos(x) \)
• \( \cos(2x) = \cos^2(x) - \sin^2(x) \) which can also be written as \( 2\cos^2(x) - 1 \) or \( 1 - 2\sin^2(x) \)

In the given integral \( \int \sin(x)\sin(2x) \, dx \), using the double-angle formula \( \sin(2x) = 2\sin(x)\cos(x) \) helps transform the integral into a format that allows straightforward substitution and integration.

By using these formulas, you significantly simplify the integral, reducing complex expressions to simpler parts that are easier and faster to integrate.