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Chapter 5: Problem 54

Integrals with \(\sin ^{2} x\) and \(\cos ^{2} x\) Evaluate the following integrals. $$\int \sin ^{2} x d x$$

Short Answer

Expert verified
Question: Evaluate the integral of \(\sin^2{x}\). Answer: The integral of \(\sin^2{x}\) is \(\frac{1}{2}x - \frac{1}{4}\sin{2x} + C\).

Step by step solution

01

Write the integral using the double angle identity

Using the identity \(\sin^2{x} = \frac{1 - \cos{2x}}{2}\), we can rewrite the given integral as: $$\int \sin^2{x} \, dx = \int \frac{1 - \cos{2x}}{2} \, dx$$
02

Distribute the constant and split the integral

Now, distribute the constant \(\frac{1}{2}\) and split the integral into two separate integrals: $$\int \frac{1 - \cos{2x}}{2} \, dx = \frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos{2x} \, dx$$
03

Integrate both terms

Integrate both terms of the split integral: $$\frac{1}{2} \int 1 \, dx = \frac{1}{2}x$$ $$\frac{1}{2} \int \cos{2x} \, dx = \frac{1}{4}\sin{2x}$$ Combine both terms and add the constant of integration, \(C\):
04

Final Answer

$$\int \sin^2{x} \, dx = \frac{1}{2}x - \frac{1}{4}\sin{2x} + C$$

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