What is the integral of 3/(2sqrt(9-x^2)) ?

The integral of 3/(2sqrt(9-x^2)) is 3/2 arcsin(1/3 x)+C

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integral of 3/(2sqrt(9-x^2))

\int \frac{3}{2 9 - x ^{2}} d x

\frac{3}{2} arcsin (\frac{1}{3} x) + C

Solution steps

\int \frac{3}{2 9 - x ^{2}} d x

Take the constant out:

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

= \frac{3}{2} \cdot \int \frac{1}{9 - x ^{2}} d x

Apply Trigonometric Substitution

: \int 1 d u

= \frac{3}{2} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{3}{2} \cdot 1 \cdot u

Substitute back

u = arcsin (\frac{1}{3} x)

= \frac{3}{2} \cdot 1 \cdot arcsin (\frac{1}{3} x)

Simplify

= \frac{3}{2} arcsin (\frac{1}{3} x)

Add a constant to the solution

= \frac{3}{2} arcsin (\frac{1}{3} x) + C

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What is the integral of 3/(2sqrt(9-x^2)) ?
The integral of 3/(2sqrt(9-x^2)) is 3/2 arcsin(1/3 x)+C