What is the integral of 1/(sqrt(-x^2+12x-32)) ?

The integral of 1/(sqrt(-x^2+12x-32)) is arcsin(1/2 (x-6))+C

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integral of 1/(sqrt(-x^2+12x-32))

\int \frac{1}{- x ^{2} + 12 x - 32} d x

arcsin (\frac{1}{2} (x - 6)) + C

Solution steps

\int \frac{1}{- x ^{2} + 12 x - 32} d x

Complete the square

- x^{2} + 12 x - 32 : - (x - 6)^{2} + 4

= \int \frac{1}{- ( x - 6 ) ^{2} + 4} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 4} d u

= \int \frac{1}{- u ^{2} + 4} d u

Apply Trigonometric Substitution

: \int 1 d v

= \int 1 d v

Integral of a constant:

\int a d x = a x

= 1 \cdot v

Substitute back

= 1 \cdot arcsin (\frac{1}{2} (x - 6))

Simplify

= arcsin (\frac{1}{2} (x - 6))

Add a constant to the solution

= arcsin (\frac{1}{2} (x - 6)) + C

\frac{\partial}{\partial x} (2 y + x e^{x y})

n = 1 \sum \infty \frac{1}{2 ^{2 n - 1}}

\int \frac{k}{x ^{6}} d x

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

\int - \frac{4000 x}{( 1 + x ^{2} )} d x

What is the integral of 1/(sqrt(-x^2+12x-32)) ?
The integral of 1/(sqrt(-x^2+12x-32)) is arcsin(1/2 (x-6))+C