Question

find integration cos^2 x/2​

Answer 1

$\underline{\texttt{Solution :}}$

$\mathrm{Now,\:\int cos^{2}\frac{x}{2}\:dx}$

$\mathrm{=\frac{1}{2} \int 2\:cos^{2}\frac{x}{2}\:dx}$

$\mathrm{=\frac{1}{2} \int (1+cosx)\:dx}$

$\mathrm{=\frac{1}{2}\int dx+\frac{1}{2}\int cosx\:dx}$

$\mathrm{=\frac{1}{2}x+\frac{1}{2}sinx+C}$

$\texttt{where C is integral constant}$

$\to \boxed{\mathrm{\int cos^{2}\frac{x}{2}\:dx=\frac{1}{2}x+\frac{1}{2}sinx+C}}$

$\texttt{which is the required integral}$

$\underline{\texttt{Mathematical Deduction :}}$

$\mathrm{cosx=cos(\frac{x}{2}+\frac{x}{2})}$

$\mathrm{=cos\frac{x}{2}\:cos\frac{x}{2}-sin\frac{x}{2}\:sin\frac{x}{2}}$

$\mathrm{=cos^{2}\frac{x}{2}-sin^{2}\frac{x}{2}}$

$\mathrm{=cos^{2}\frac{x}{2}-(1-cos^{2}\frac{x}{2})}$

$\boxed{\mathrm{since,\:sin^{2}\frac{x}{2}+cos^{2}\frac{x}{2}=1}}$

$\mathrm{=cos^{2}\frac{x}{2}-1+cos^{2}\frac{x}{2}}$

$\mathrm{=2\:cos^{2}\frac{x}{2}-1}$

$\to \mathrm{2\:cos^{2}\frac{x}{2}=1+cosx}$