What is the integration of sin(x) cos(x)?

Understand the Problem

The question is asking for the integral of the product of sine and cosine functions. The approach involves using trigonometric identities or integration techniques to find the antiderivative of sin(x) * cos(x).

$-\frac{1}{4} \cos(2x) + C'$

The integral of sin(x) cos(x) is $-\frac{1}{4} \cos(2x) + C'$

Steps to Solve

1. Use a trigonometric identity to simplify the integrand

Use the identity $\sin(x) \cos(x) = \frac{1}{2} \sin(2x)$ to simplify the integral:

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

1. Simplify the integral

Factor out the constant $\frac{1}{2}$:

$$\int \frac{1}{2} \sin(2x) , dx = \frac{1}{2} \int \sin(2x) , dx$$

1. Integrate the simplified function

Recall the antiderivative of $\sin(2x)$:

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

1. Combine the results

Combine the constant and the antiderivative:

$$\frac{1}{2} \left( -\frac{1}{2} \cos(2x) + C \right) = -\frac{1}{4} \cos(2x) + \frac{C}{2}$$

Since any constant, when multiplied by another constant, remains a constant, we can simplify this as:

$$-\frac{1}{4} \cos(2x) + C'$$

The integral of sin(x) cos(x) is $-\frac{1}{4} \cos(2x) + C'$