Integrals of the Form

$\int \sin^n x \cos^m x \, dx$

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🔍 Step 0: Identify the Powers

Look at the exponents:

• Is one of them odd? → use substitution
• Are both even? → use identities

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🟢 Case 1: At Least One Power is Odd

💡 Core Idea

Save one factor and convert the rest using identities.

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👉 If $\sin x$ is odd:

• Save one $\sin x$
• Use: $\sin^2 x = 1 - \cos^2 x$
• Substitute: $u = \cos x$

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👉 If $\cos x$ is odd:

• Save one $\cos x$
• Use: $\cos^2 x = 1 - \sin^2 x$
• Substitute: $u = \sin x$

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✏️ Example 1

Evaluate:

$$\int \sin^3 x \cos^2 x \, dx$$

Step 1: Separate the odd power

$$\sin^3 x = \sin^2 x \cdot \sin x$$

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Step 2: Use identity

$$= \int (1 - \cos^2 x)\cos^2 x \sin x \, dx$$

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Step 3: Substitution

Let:

• $u = \cos x$
• $du = -\sin x \, dx$

$$= -\int (1 - u^2)u^2 \, du$$

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Step 4: Expand and integrate

$$= -\int (u^2 - u^4)\,du$$ $$= -\left(\frac{u^3}{3} - \frac{u^5}{5}\right) + C$$

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✅ Final Answer

$$= -\frac{\cos^3 x}{3} + \frac{\cos^5 x}{5} + C$$

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🔵 Case 2: Both Powers are Even

💡 Core Idea

Use power-reducing identities to simplify.

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Key Identities

$$\sin^2 x = \frac{1 - \cos(2x)}{2}, \quad \cos^2 x = \frac{1 + \cos(2x)}{2}$$

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✏️ Example 2

Evaluate:

$$\int \sin^2 x \cos^2 x \, dx$$

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Step 1: Apply identities

$$= \int \frac{(1 - \cos(2x))(1 + \cos(2x))}{4} \, dx$$

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Step 2: Simplify

$$= \int \frac{1 - \cos^2(2x)}{4} \, dx = \int \frac{\sin^2(2x)}{4} \, dx$$

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Step 3: Apply identity again

$$\sin^2(2x) = \frac{1 - \cos(4x)}{2}$$ $$= \int \frac{1 - \cos(4x)}{8} \, dx$$

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Step 4: Integrate

$$= \frac{1}{8}\left(x - \frac{\sin(4x)}{4}\right) + C$$

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🧠 Final Summary

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🚀 Key Takeaway

Always aim to reduce the integral into:

• a polynomial, or
• a basic trig integral you already know