Problem 57 - Thomas Calculus

Question

Use reduction formulas to evaluate the integrals. \(\int \sin ^{2} 2 \theta \cos ^{3} 2 \theta d \theta\)

Step-by-Step Solution

Answer

Apply reduction formula and substitution to solve.

Step 1: Identify reduction formula

For integrals involving powers of sine and cosine such as \( \int \sin^m x \cos^n x \ dx \), a useful reduction formula applies. We can apply: \( \int \sin^m x \cos^n x \ dx = \frac{\sin^{m-1} x \cos^{n+1} x}{m+n} + \frac{m - 1}{m+n} \int \sin^{m-2} x \cos^n x \ dx \) when \( n \) is odd.

Step 2: Set up the integral with substitution

Let \( u = 2\theta \), then \( du = 2 d\theta \) or \( d\theta = \frac{1}{2} du \). The integral becomes \( \int \sin^2 u \cos^3 u \ \frac{1}{2} du = \frac{1}{2} \int \sin^2 u \cos^3 u \ du \).

Step 3: Express integral using reduction formula

Since \( n = 3 \), the integral \( \int \sin^2 u \cos^3 u \ du \) achieves a first reduction to \( \frac{\sin^3 u \cos^2 u}{5} + \frac{2}{5} \int \sin^4 u \cos u \ du \).

Step 4: Further reduce remaining integral

For the integral \( \int \sin^4 u \cos u \ du \), apply integration by parts or a reduction pathway for sine powers. Apply changes until integral reaches solvable form.

Step 5: Back substitution

After evaluating the integrals, replace \( u \) back with \( 2\theta \). The final result is simplified to be in terms of \( \theta \).

Step 6: Final calculation and cleanup

Ensure simplification of trigonometric identities and combine constants. Check all calculations to ensure the final answer is reduced and correct.

Key Concepts

Integration by Parts

Integration by parts is a technique used to evaluate integrals and is particularly useful when dealing with the product of two functions. The idea is derived from the product rule of differentiation. The formula is given by:

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

When applying integration by parts, identify two parts of the integral - one to differentiate (\( u \)) and one to integrate (\( dv \)). Then, carefully perform the differentiation and integration separately. This method systematically "breaks down" the integral to make it easier to solve. It is especially handy in exercises where substitution methods are insufficient. For instance, in reducing integrals involving powers of sine and cosine, sometimes a combination of integration by parts and other techniques like substitution can help to simplify the process.

Trigonometric Integration

Trigonometric integration deals with integrals of trigonometric functions. These integrals are tackled using specific strategies and identities that simplify them. When you encounter a product of trigonometric functions such as sine and cosine, consider:

• When both functions have the same power, reducing powers using trigonometric identities can simplify the problem.
• If one of the powers is odd, factor a single sine or cosine and use it for substitution.

The goal is to convert the integral into a form that is straightforward to solve. Often, integrals involving high powers of sine and cosine benefit from employing reduction formulas or by transforming them using identities. The process involves breaking down the integral step-by-step, simplifying as much as possible, until reaching an easier form that can be integrated directly.

Substitution Method

Substitution is a fundamental technique in integral calculus that simplifies an integral by substituting a part of the expression with a single variable. This method is akin to reversing the chain rule for derivatives. Typically, you want to:

• Choose a substitution that simplifies the integral. Usually, set \( u \) equal to a function within the integral.
• Determine the differential \( du = g'(x) \, dx \) to replace \( dx \).

This approach transforms the original integral into a new form that is easier to evaluate. The substitution is particularly effective when the integral involves composite functions or when a trigonometric integral requires simplifying transformations. Once solved, substitute back your original variable to express the solution in terms of the original integration variable.

Trigonometric Identities

Trigonometric identities are pivotal tools in calculus and algebra that relate the angles and sides of triangles. These identities simplify complex trigonometric expressions and are indispensable in solving integrals involving trigonometric functions.
Some crucial identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Double Angle Formulas: \( \sin 2x = 2 \sin x \cos x \) and \( \cos 2x = \cos^2 x - \sin^2 x \)

These identities come in handy to reduce the powers of sine and cosine to a more manageable form. They also allow expressing functions in such a way that direct integration becomes feasible. Utilizing these identities wisely can transform complex integrals into a simpler form, helping in both the setup and calculation process during integration.