91Ó°ÊÓ

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

Itis a methodto obtain antiderivatives. Generally, formula for $u-$substitution is:

$∫f\left(g\right(x\left)\right)g\text{'}\left(x\right)dx=∫f\left(u\right)du$

Here, $u=g\left(x\right);du=g\text{'}\left(x\right)dx$.

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:

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

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

After applying u- substitution method, we get:

role="math" localid="1653045028107" $$\begin{gathered} 2.∫\sin \mathrm{θ}\cos \mathrm{θ}d\mathrm{θ}=2.∫udu \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}=2.\frac{{\mathit{u}}^{\mathit{2}}}{\mathit{2}}[\mathrm{Applying}\mathrm{Power}\mathrm{Rule}] \end{gathered}$$

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