# Integral of $\sin{\left(x \right)} \cos{\left(x \right)}$

The calculator will find the integral/antiderivative of $\sin{\left(x \right)} \cos{\left(x \right)}$, with steps shown.

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Find $\int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$.

### Solution

Let $u=\sin{\left(x \right)}$.

Then $du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$ (steps can be seen here), and we have that $\cos{\left(x \right)} dx = du$.

Therefore,

$$\color{red}{\int{\sin{\left(x \right)} \cos{\left(x \right)} d x}} = \color{red}{\int{u d u}}$$

Apply the power rule $\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$ $\left(n \neq -1 \right)$ with $n=1$:

$$\color{red}{\int{u d u}}=\color{red}{\frac{u^{1 + 1}}{1 + 1}}=\color{red}{\left(\frac{u^{2}}{2}\right)}$$

Recall that $u=\sin{\left(x \right)}$:

$$\frac{\color{red}{u}^{2}}{2} = \frac{\color{red}{\sin{\left(x \right)}}^{2}}{2}$$

Therefore,

$$\int{\sin{\left(x \right)} \cos{\left(x \right)} d x} = \frac{\sin^{2}{\left(x \right)}}{2}$$

$$\int{\sin{\left(x \right)} \cos{\left(x \right)} d x} = \frac{\sin^{2}{\left(x \right)}}{2}+C$$
Answer: $\int{\sin{\left(x \right)} \cos{\left(x \right)} d x}=\frac{\sin^{2}{\left(x \right)}}{2}+C$