What is the integral of ((6x+6))/(sqrt(x^2+2x-4)) ?

The integral of ((6x+6))/(sqrt(x^2+2x-4)) is 6sqrt(x^2+2x-4)+C

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integral of ((6x+6))/(sqrt(x^2+2x-4))

\int \frac{( 6 x + 6 )}{x ^{2} + 2 x - 4} d x

6 x^{2} + 2 x - 4 + C

Solution steps

\int \frac{6 x + 6}{x ^{2} + 2 x - 4} d x

Apply u-substitution

: \int \frac{3}{u} d u

= \int \frac{3}{u} d u

Take the constant out:

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

= 3 \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 3 \cdot 2 u^{\frac{1}{2}}

Substitute back

u = x^{2} + 2 x - 4

= 3 \cdot 2 (x^{2} + 2 x - 4)^{\frac{1}{2}}

Simplify

3 \cdot 2 (x^{2} + 2 x - 4)^{\frac{1}{2}} : 6 x^{2} + 2 x - 4

= 6 x^{2} + 2 x - 4

Add a constant to the solution

= 6 x^{2} + 2 x - 4 + C

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What is the integral of ((6x+6))/(sqrt(x^2+2x-4)) ?
The integral of ((6x+6))/(sqrt(x^2+2x-4)) is 6sqrt(x^2+2x-4)+C