Question

Integrate $\int \sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\mathrm{d}x$.

Answer 1

Calculate the given expression:

Given, $\int \sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\mathrm{d}x$

Let, $\mathrm{I}=\int \sin \left(x\right)\cos \left(x\right)\mathrm{d}x$

Let, $\sin \left(x\right)=t$

Differentiating both sides with respect to $x$,

$\begin{array}{cc}& \cos \left(x\right)=\frac{\mathrm{d}t}{\mathrm{d}x}\\ \Rightarrow & \cos \left(x\right)\mathrm{d}x=\mathrm{d}t\end{array}$

$\begin{array}{ccc}\Rightarrow & \mathrm{I}=\int t\mathrm{d}t& \left[\because sin\left(x\right)=t,cos\left(x\right)dx=dt\right]\\ \Rightarrow & \mathrm{I}=\frac{t^2}{2}+\mathrm{C}& \left[\because \int x^n=\frac{x^{n+1}}{n+1}\right]\\ \Rightarrow & \mathrm{I}=\frac{{\sin }^{2}x}{2}+\mathrm{C}& \end{array}$

Therefore,$\mathbf{\int }\mathbf{sin}\left(x\right)\mathbf{cos}\left(x\right)\mathbf{dx}\mathbf{=}\frac{{\mathbf{sin}}^{\mathbf{2}}\left(x\right)}{\mathbf{2}}\mathbf{+}\mathbf{C}$.