What is the integral of 1/(sqrt(-t^2+2t+8)) ?

The integral of 1/(sqrt(-t^2+2t+8)) is arcsin(1/3 (t-1))+C

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integral of 1/(sqrt(-t^2+2t+8))

\int \frac{1}{- t ^{2} + 2 t + 8} d t

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

Solution steps

\int \frac{1}{- t ^{2} + 2 t + 8} d t

Complete the square

- t^{2} + 2 t + 8 : - (t - 1)^{2} + 9

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

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} (t - 1))

Simplify

= arcsin (\frac{1}{3} (t - 1))

Add a constant to the solution

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

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What is the integral of 1/(sqrt(-t^2+2t+8)) ?
The integral of 1/(sqrt(-t^2+2t+8)) is arcsin(1/3 (t-1))+C