Question

Integration of sin x cos x

Answer 1

Dear Student

$$\begin{gathered} Lets\mathrm{I}=\int \sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)d\mathrm{x} \\ \left[\because \mathrm{we}\mathrm{know},\sin 2\mathrm{\theta }=2.\mathrm{sin\theta }.\mathrm{cos\theta }\right] \\ \Rightarrow \mathrm{I}=\frac{1}{2}\int 2\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)d\mathrm{x} \\ =\frac{1}{2}\int \sin \left(2\mathrm{x}\right)d\mathrm{x} \\ =\frac{1}{2}\left(-\frac{\cos \left(2\mathrm{x}\right)}{2}\right)+\mathrm{C}=-\frac{\cos \left(2\mathrm{x}\right)}{4}+\mathrm{C} \\ \mathrm{Where},\mathrm{C}\mathrm{is}\mathrm{constant}. \\ \mathrm{Hence},\int \sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)d\mathrm{x}=-\frac{\cos \left(2\mathrm{x}\right)}{4}+\mathrm{C} \end{gathered}$$
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