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

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

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integral of 1/(sqrt(-x^2+2x+3))

\int \frac{1}{- x ^{2} + 2 x + 3} d x

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

Solution steps

\int \frac{1}{- x ^{2} + 2 x + 3} d x

Complete the square

- x^{2} + 2 x + 3 : - (x - 1)^{2} + 4

= \int \frac{1}{- ( x - 1 ) ^{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 - 1))

Simplify

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

Add a constant to the solution

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

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

\frac{3}{4 x + 1}

\frac{d}{d x} (sin^{2} (2) x^{2})

\int \frac{5}{1 + 4 x ^{2}} d x

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