Question

A Preliminary Example: Integrating \(\int \cos j x \sin ^{k} x d x\) Where \(k\) is Odd Evaluate \(\int \cos ^{2} x \sin ^{3} x d x\).

Short answer

The integral evaluates to \(-\frac{\cos^3 x}{3} + \frac{\cos^5 x}{5} + C\)."

Step-by-step solution

Identify Trigonometric Identity

Since we know that \(k = 3\) is odd, we recognize that simplifying using a trigonometric identity will make integration easier. We can use the identity \( \sin^2 x = 1 - \cos^2 x \) to simplify the expression.

Rewrite Integral Using Identity

Rewrite \( \sin^3 x \) as \( \sin x \cdot \sin^2 x \). Replace \( \sin^2 x \) with \( 1 - \cos^2 x \) to get: \[ \int \cos^2 x \sin x (1 - \cos^2 x) \, dx \]

Expand the Expression

Next, distribute the \( \sin x \) across the expression to get:\[ \int \sin x \cos^2 x \, dx - \int \sin x \cos^4 x \, dx \]

Use Substitution Method

Let \( u = \cos x \), which implies \( du = -\sin x \, dx \). Rewrite the integrals using this substitution:\[ -\int u^2 \, du + \int u^4 \, du \]

Integrate Each Term

Integrate each term separately:\[ -\int u^2 \, du = -\frac{u^3}{3} = -\frac{\cos^3 x}{3} \]\[ \int u^4 \, du = \frac{u^5}{5} = \frac{\cos^5 x}{5} \]

Combine Solutions

Combine both integrated results from the previous step:\[ -\frac{\cos^3 x}{3} + \frac{\cos^5 x}{5} + C \]Here, \( C \) is the constant of integration.

Key concepts

Substitution Method

The substitution method is a powerful tool in integration, especially for more complex integrals involving trigonometric functions. This technique involves changing the variable of integration to simplify the integral.
This helps to make the integral more manageable. The core idea is to replace a part of the integrand with a new variable that simplifies the expression.

Consider our integration task: \( \int \cos^2 x \sin^3 x \, dx \). By making a substitution, like \( u = \cos x \), you gain two advantages:

• The integrand translates to a power of \( u \) which is simpler to work with.
• The differential \( du = -\sin x \, dx \) helps in direct substitution.

This converts the original integral into one involving only \( u \), allowing algebraic sequencing for easier integration. The substitution evolves the complexity of trigonometric integrals, turning them into polynomial forms whenever feasible.
Once this transformation is complete, it mechanicalizes the process of integrating what would otherwise be a daunting trigonometric challenge.

Trigonometric Identities

Trigonometric identities are invaluable in transforming and simplifying expressions before integration. They are particularly helpful with integrals involving sine and cosine raised to powers.

For instance, in the problem \( \int \cos^2 x \sin^3 x \, dx \), the identity \( \sin^2 x = 1 - \cos^2 x \) is used.

• This identity helps in expressing powers of \( \sin x \) in terms of \( \cos x \).
• It streamlines expressions that contain composite trigonometric powers.

Using this identity, \( \sin^3 x \) can be rewritten as \( \sin x (1 - \cos^2 x) \), which simplifies the integration process.
This step is critical as it prepares the integrand for the substitution method and is a strategic prelude to easy calculus manipulations.
Trigonometric identities bridge complex integrations and present paths to substitution and solution.

Integration Techniques

Integration techniques encompass a range of strategies to solve integrals, creating a repertoire of methods to tackle different forms of integrands. Each integral type might require a unique approach, and uncovering the right method is often the challenge.

Consider separate strategies like substitution, partial fractions, or trigonometric integrals:

• Substitution: Important for integrals involving repetitive functions that transform into polynomials.
• Trigonometric Techniques: Useful for formats involving products or powers of sine and cosine, employing identities to simplify.

In the example \( \int \cos^2 x \sin^3 x \, dx \), substitution and trigonometric identities were crucial. They collectively assisted in breaking the integral into simpler chunks that, individually, were manageable to solve.
This integrated approach ensures that every facet of an integral is systematically catered to, underscoring how robust understanding of these techniques bolsters effective problem-solving.