Question

How to do the integration of two functions multiplied to each other using the formula? Example-integration of sin x cos x dx

Answer 1

$$\begin{gathered} \int \mathrm{sinxcosxdx} \\ \mathrm{Taking}\mathrm{sinx}\mathrm{as}\mathrm{first}\mathrm{function}\mathrm{and}\cos \mathrm{x}\mathrm{as}\mathrm{second}\mathrm{function}. \\ =\mathrm{sinx}\int \mathrm{cosxdx}-\int \left(\left(\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{sinx}\right)\int \mathrm{cosxdx}\right)\mathrm{dx} \\ ={\sin }^2\mathrm{x}-\int \mathrm{cosxsinxdx} \\ \int \mathrm{sinxcosxdx}={\sin }^2\mathrm{x}-\int \mathrm{cosxsinxdx} \\ 2\int \mathrm{cosxsinxdx}={\sin }^2\mathrm{x} \\ \int \mathrm{cosxsinxdx}=\int \mathrm{sinxcosxdx}=\frac{{\sin }^2\mathrm{x}}{2}+\mathrm{C} \\ \mathrm{We}\mathrm{have}\mathrm{used}\mathrm{the}\mathrm{formula},\int \mathrm{cosxdx}=\mathrm{sinx}+\mathrm{C} \\ \int \mathrm{sinxdx}=-\mathrm{cosx}+\mathrm{C} \end{gathered}$$