What is the integral from 0 to 2 of 2/(sqrt(6-x^2)) ?

The integral from 0 to 2 of 2/(sqrt(6-x^2)) is 2arcsin(sqrt(2/3))

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integral from 0 to 2 of 2/(sqrt(6-x^2))

\int_{0}^{2} \frac{2}{6 - x ^{2}} d x

2 arcsin (\frac{2}{3})

Decimal

1.91063 \dots

Solution steps

\int_{0}^{2} \frac{2}{6 - x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{2} \frac{1}{6 - x ^{2}} d x

Apply Trigonometric Substitution

= 2 \cdot \int_{0}^{a r c s i n (\frac{2}{3})} 1 d u

Integral of a constant:

\int a d x = a x

= 2 [1 \cdot u]_{0}^{a r c s i n (\frac{2}{3})}

Simplify

= 2 [u]_{0}^{a r c s i n (\frac{2}{3})}

Compute the boundaries:

arcsin (\frac{2}{3})

= 2 arcsin (\frac{2}{3})

What is the integral from 0 to 2 of 2/(sqrt(6-x^2)) ?
The integral from 0 to 2 of 2/(sqrt(6-x^2)) is 2arcsin(sqrt(2/3))