What is the integral of (ax+b)/(cx+ ?

The integral of (ax+b)/(cx+ is a 1/(c^2)(cx+d-dln|cx+d|)+b 1/c ln|cx+d|+C

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integral of (ax+b)/(cx+

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

a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣) + b \frac{1}{c} ln ∣ c x + d ∣ + C

Solution steps

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

Expand

\frac{a x + b}{c x + d} : \frac{a x}{c x + d} + \frac{b}{c x + d}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{a x}{c x + d} d x + \int \frac{b}{c x + d} d x

\int \frac{a x}{c x + d} d x = a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣)

\int \frac{b}{c x + d} d x = b \frac{1}{c} ln ∣ c x + d ∣

= a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣) + b \frac{1}{c} ln ∣ c x + d ∣

Add a constant to the solution

= a \frac{1}{c ^{2}} (c x + d - d ln ∣ c x + d ∣) + b \frac{1}{c} ln ∣ c x + d ∣ + C

\frac{d}{d x} (\frac{4 x ^{\frac{3}{2}}}{x})

\int 10 x - 3 x^{2} - x^{3} d x

x (\frac{d y}{d x}) + 2 y = 0

\int (x^{2} - x) cos (x) d x

x \to 5 - lim (\frac{2}{( x - 5 )})

What is the integral of (ax+b)/(cx+ ?
The integral of (ax+b)/(cx+ is a 1/(c^2)(cx+d-dln|cx+d|)+b 1/c ln|cx+d|+C