Question

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

Answer 1

Evaluate the given Integration:

Let,

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

Hence $\int {\cos }^2\left(\mathrm{x}\right)\mathrm{dx}$ $=\frac{\mathrm{x}}{2}+\frac{\sin \left(2\mathrm{x}\right)}{4}+\mathrm{C}$.