What is the integral of 1/(sqrt(-4(x+5/2)^2+4)) ?

The integral of 1/(sqrt(-4(x+5/2)^2+4)) is 1/2 arcsin(x+5/2)+C

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integral of 1/(sqrt(-4(x+5/2)^2+4))

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

\frac{1}{2} arcsin (x + \frac{5}{2}) + C

Solution steps

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

- 4 (x + \frac{5}{2})^{2} + 4 = - 4 x^{2} - 20 x - 21

= \int \frac{1}{- 4 x ^{2} - 20 x - 21} d x

Complete the square

- 4 x^{2} - 20 x - 21 : - 4 (x + \frac{5}{2})^{2} + 4

= \int \frac{1}{- 4 ( x + \frac{5}{2} ) ^{2} + 4} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = arcsin (u)

= \frac{1}{2} arcsin (u)

Substitute back

u = x + \frac{5}{2}

= \frac{1}{2} arcsin (x + \frac{5}{2})

Add a constant to the solution

= \frac{1}{2} arcsin (x + \frac{5}{2}) + C

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What is the integral of 1/(sqrt(-4(x+5/2)^2+4)) ?
The integral of 1/(sqrt(-4(x+5/2)^2+4)) is 1/2 arcsin(x+5/2)+C