# Use geometry and properties of integrals to evaluate the following definite inte

Use geometry and properties of integrals to evaluate the following definite integrals.
${\int }_{4}^{0}\left(2x+\sqrt{16-{x}^{2}}\right)dx$
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Given:

Let ${x}^{2}=u$
Then
With these substitutions, the integral in (1) transforms to
$I=\frac{1}{2}\mathrm{sin}\left(u\right){\mathrm{cos}}^{8}\left(u\right)du$ (2)
Now the integrand in (2)
Suggests that cosine function should be used for change of variable,
Since its derivative sine function is present in the integrand. Thus
Let $\mathrm{cos}\left(u\right)=v$
Then $-\mathrm{sin}\left(u\right)du=dv⇒\mathrm{sin}\left(u\right)du=-dv$
Therefore
$I=-\frac{1}{2}\int {v}^{8}dv$ (3)
now,
We know that for $n\ne -1,\frac{{x}^{n+1}}{n+1}$ is an antiderivative of the function $f\left(x\right)={x}^{n}$
So that
$I=-\frac{1}{2}\left[\frac{{v}^{9}}{9}\right]+C$
$=-\frac{1}{18}{\mathrm{cos}}^{9}\left(u\right)+C$
$=-\frac{1}{18}{\mathrm{cos}}^{9}\left({x}^{2}\right)+C$