What is the integral of x/(3x+5) ?

The integral of x/(3x+5) is 1/3 x-5/9 ln|3x+5|+C

Popular >

integral of x/(3x+5)

\int \frac{x}{3 x + 5} d x

\frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣ + C

Solution steps

\int \frac{x}{3 x + 5} d x

Long division

\frac{x}{3 x + 5} : \frac{1}{3} - \frac{5}{3 ( 3 x + 5 )}

= \int \frac{1}{3} - \frac{5}{3 ( 3 x + 5 )} d x

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{1}{3} d x - \int \frac{5}{3 ( 3 x + 5 )} d x

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{5}{3 ( 3 x + 5 )} d x = \frac{5}{9} ln ∣ 3 x + 5 ∣

= \frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣

Add a constant to the solution

= \frac{1}{3} x - \frac{5}{9} ln ∣ 3 x + 5 ∣ + C

\int x^{3} (x + 7)^{9} d x

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

\int \frac{( x + 1 )}{( x ^{3} + x ^{2} - 6 x )} d x

\int 2 cos (8 x) d x

\int sin^{5} (m) y d y

What is the integral of x/(3x+5) ?
The integral of x/(3x+5) is 1/3 x-5/9 ln|3x+5|+C