What is the integral of 1/(sqrt(15+4x-4x^2)) ?

The integral of 1/(sqrt(15+4x-4x^2)) is 1/2 arcsin(1/2 (x-1/2))+C

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integral of 1/(sqrt(15+4x-4x^2))

\int \frac{1}{15 + 4 x - 4 x ^{2}} d x

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

Solution steps

\int \frac{1}{15 + 4 x - 4 x ^{2}} d x

Complete the square

15 + 4 x - 4 x^{2} : - 4 (x - \frac{1}{2})^{2} + 16

= \int \frac{1}{- 4 ( x - \frac{1}{2} ) ^{2} + 16} d x

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 1 d v

= \frac{1}{2} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot v

Substitute back

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

Simplify

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

Add a constant to the solution

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

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What is the integral of 1/(sqrt(15+4x-4x^2)) ?
The integral of 1/(sqrt(15+4x-4x^2)) is 1/2 arcsin(1/2 (x-1/2))+C