Question

Integrate $\int {\sin }^2\left(\mathrm{x}\right)\mathrm{dx}$.

Answer 1

Solve the given integral function by integration

Let, $\mathrm{I}=\int {\sin }^2\left(\mathrm{x}\right)\mathrm{dx}$

$$\begin{gathered} \int {\sin }^2\left(\mathrm{x}\right)\mathrm{dx}=\int \frac{1-\cos \left(2\mathrm{x}\right)}{2}dx\left[\because \cos \left(2\mathrm{x}\right)=1-2{\sin }^2\left(\mathrm{x}\right)\right] \\ \Rightarrow \mathrm{I}=\int \frac{\mathrm{dx}}{2}-\frac{1}{2}\int \cos \left(2\mathrm{x}\right)\mathrm{dx}\left[\because \int \mathrm{dx}=\mathrm{x}+\mathrm{C},\int \cos \left(\mathrm{cx}\right)=\frac{\sin \left(\mathrm{cx}\right)}{\mathrm{c}}+\mathrm{C}\right] \\ \Rightarrow \mathrm{I}=\frac{\mathrm{x}}{2}-\frac{\sin \left(2\mathrm{x}\right)}{4}+\mathrm{C} \end{gathered}$$

Hence, $\int {\sin }^2\left(\mathrm{x}\right)\mathrm{dx}$ is integrated as $\frac{\mathrm{x}}{2}-\frac{\sin \left(2\mathrm{x}\right)}{4}+\mathrm{C}$.