\[\int \frac{2}{\sqrt{25-{x}^{2}}} \, dx\]
1
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
2∫125−x2 dx2\int \frac{1}{\sqrt{25-{x}^{2}}} \, dx

2
Use Trigonometric Substitution
Let x=5sin⁡ux=5\sin{u}, dx=5cos⁡u dudx=5\cos{u} \, du

3
Substitute variables from above.
∫125−(5sin⁡u)2×5cos⁡u du\int \frac{1}{\sqrt{25-{(5\sin{u})}^{2}}}\times 5\cos{u} \, du

4
Simplify.
∫1 du\int 1 \, du

5
Use this rule: ∫a dx=ax+C\int a \, dx=ax+C.
uu

6
From the earlier steps, we know that:
u=sin⁡−1(15x)u=\sin^{-1}{(\frac{1}{5}x)}

7
Substitute the above back into the original integral.
sin⁡−1(15x)\sin^{-1}{(\frac{1}{5}x)}

8
Rewrite the integral with the completed substitution.
2sin⁡−1(x5)2\sin^{-1}{(\frac{x}{5})}

9
Add constant.
2sin⁡−1(x5)+C2\sin^{-1}{(\frac{x}{5})}+C

Done
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