What is the integral of (30)/(sqrt(45+12x-x^2)) ?

The integral of (30)/(sqrt(45+12x-x^2)) is 30arcsin(1/9 (x-6))+C

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integral of (30)/(sqrt(45+12x-x^2))

\int \frac{30}{45 + 12 x - x ^{2}} d x

30 arcsin (\frac{1}{9} (x - 6)) + C

Solution steps

\int \frac{30}{45 + 12 x - x ^{2}} d x

Take the constant out:

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

= 30 \cdot \int \frac{1}{45 + 12 x - x ^{2}} d x

Complete the square

45 + 12 x - x^{2} : - (x - 6)^{2} + 81

= 30 \cdot \int \frac{1}{- ( x - 6 ) ^{2} + 81} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 81} d u

= 30 \cdot \int \frac{1}{- u ^{2} + 81} d u

Apply Trigonometric Substitution

: \int 1 d v

= 30 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 30 \cdot 1 \cdot v

Substitute back

= 30 \cdot 1 \cdot arcsin (\frac{1}{9} (x - 6))

Simplify

= 30 arcsin (\frac{1}{9} (x - 6))

Add a constant to the solution

= 30 arcsin (\frac{1}{9} (x - 6)) + C

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What is the integral of (30)/(sqrt(45+12x-x^2)) ?
The integral of (30)/(sqrt(45+12x-x^2)) is 30arcsin(1/9 (x-6))+C