91Ó°ÊÓ

The u-substitution reserves the chain rule and it is used to solve integrals.

It is a method to obtain antiderivatives.Generally, formula for u-substitution is:

$\mathbf{∫}\mathit{f}\left(g\left(x\right)\right)\mathit{g}\mathbf{\text{'}}\left(x\right)\mathit{d}\mathit{x}\mathbf{=}\mathbf{∫}\mathit{f}\left(u\right)\mathit{d}\mathit{u}$

Here, $\mathit{u}\mathbf{=}\mathit{g}\left(x\right)\mathbf{;}\mathit{d}\mathit{u}\mathbf{=}\mathit{g}\mathbf{\text{'}}\left(x\right)\mathit{d}\mathit{x}$.

The given trigonometric integral is $∫2 \sin \mathrm{θ} \cos \mathrm{θ} d\mathrm{θ}$.

Take the constant out by using the property $∫a·f\left(x\right)dx=a·∫f\left(x\right)dx$.

So, after using the above formula in the trigonometric identity, we get:

$∫2 \sin \mathrm{θ} \cos \mathrm{θ} d\mathrm{θ}=2 ·∫\sin \left(\mathrm{θ}\right) \cos \left(\mathrm{θ}\right) d\mathrm{θ}$

Let $u=\sin \mathrm{θ}$then.$du=\cos \mathrm{θ}d\mathrm{θ}$

After applying u-substitution method, we get:

$$\begin{gathered} 2 ·∫\sin \mathrm{θ} \cos \mathrm{θ} d\mathrm{θ}=2 ·∫udu \\ =2 ·\frac{u^2}{2}\left[\text{Applying Power Rule}\right] \\ =u^2 \end{gathered}$$

Now, substituting back$u=\sin \mathrm{θ}$ , we get:

$u={\sin }^2\mathrm{θ}$

On further simplification:

$2 ·\frac{{\sin }^2\mathrm{θ}}{2} ={\sin }^2\mathrm{θ}$

Hence, $∫2 \sin \mathrm{θ} \cos \mathrm{θ}d\mathrm{θ}={\sin }^2\mathrm{θ}$.