What is the integral of 1/(sqrt(-x^2+10x-16)) ?

The integral of 1/(sqrt(-x^2+10x-16)) is arcsin(1/3 (x-5))+C

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integral of 1/(sqrt(-x^2+10x-16))

\int \frac{1}{- x ^{2} + 10 x - 16} d x

arcsin (\frac{1}{3} (x - 5)) + C

Solution steps

\int \frac{1}{- x ^{2} + 10 x - 16} d x

Complete the square

- x^{2} + 10 x - 16 : - (x - 5)^{2} + 9

= \int \frac{1}{- ( x - 5 ) ^{2} + 9} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 9} d u

= \int \frac{1}{- u ^{2} + 9} 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}{3} (x - 5))

Simplify

= arcsin (\frac{1}{3} (x - 5))

Add a constant to the solution

= arcsin (\frac{1}{3} (x - 5)) + C

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What is the integral of 1/(sqrt(-x^2+10x-16)) ?
The integral of 1/(sqrt(-x^2+10x-16)) is arcsin(1/3 (x-5))+C