What is the integral of (2ax+b)/(sqrt(ax^2+bx+c)) ?

The integral of (2ax+b)/(sqrt(ax^2+bx+c)) is 2sqrt(ax^2+bx+c)+C

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integral of (2ax+b)/(sqrt(ax^2+bx+c))

\int \frac{2 a x + b}{a x ^{2} + b x + c} d x

2 a x^{2} + b x + c + C

Solution steps

\int \frac{2 a x + b}{a x ^{2} + b x + c} d x

Apply u-substitution

: \int \frac{1}{u} d u

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

Apply the Power Rule

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

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

Substitute back

u = a x^{2} + b x + c

= 2 (a x^{2} + b x + c)^{\frac{1}{2}}

Apply radical rule:

a^{\frac{1}{n}} = n a

= 2 a x^{2} + b x + c

Add a constant to the solution

= 2 a x^{2} + b x + c + C

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What is the integral of (2ax+b)/(sqrt(ax^2+bx+c)) ?
The integral of (2ax+b)/(sqrt(ax^2+bx+c)) is 2sqrt(ax^2+bx+c)+C