What is the integral of 1/(sqrt((x+2)(3-x))) ?

The integral of 1/(sqrt((x+2)(3-x))) is 2arcsin(1/(sqrt(5))sqrt(x+2))+C

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integral of 1/(sqrt((x+2)(3-x)))

\int \frac{1}{( x + 2 ) ( 3 - x )} d x

2 arcsin (\frac{1}{5} x + 2) + C

Solution steps

\int \frac{1}{( x + 2 ) ( 3 - x )} d x

Apply u-substitution

: \int \frac{1}{u ( - u + 5 )} d u

= \int \frac{1}{u ( - u + 5 )} d u

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

u (- u + 5) = u (- u + 5)

= \int \frac{1}{u - u + 5} d u

Apply u-substitution

: \int \frac{2}{- v ^{2} + 5} d v

= \int \frac{2}{- v ^{2} + 5} d v

Take the constant out:

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

= 2 \cdot \int \frac{1}{- v ^{2} + 5} d v

Apply Trigonometric Substitution

: \int 1 d w

= 2 \cdot \int 1 d w

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot w

Substitute back

= 2 \cdot 1 \cdot arcsin (\frac{1}{5} x + 2)

Simplify

= 2 arcsin (\frac{1}{5} x + 2)

Add a constant to the solution

= 2 arcsin (\frac{1}{5} x + 2) + C

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What is the integral of 1/(sqrt((x+2)(3-x))) ?
The integral of 1/(sqrt((x+2)(3-x))) is 2arcsin(1/(sqrt(5))sqrt(x+2))+C