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{7}{2 x ^{2}}

x \to \infty lim (arctan (x^{6} - x^{9}))

y = arctan (3 x)

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

\int \frac{x ^{2}}{( 5 - x ^{2} ) ^{3}} d x

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