What is the integral of ((2x))/((4+3x)^2) ?

The integral of ((2x))/((4+3x)^2) is 2(-x/(3(4+3x))+1/9 ln|4+3x|)+C

integral of ((2x))/((4+3x)^2)

integral

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

2 (- \frac{x}{3 ( 4 + 3 x )} + \frac{1}{9} ln ∣ 4 + 3 x ∣) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{x}{( 4 + 3 x ) ^{2}} d x

Apply Integration By Parts

= 2 (- \frac{x}{3 ( 4 + 3 x )} - \int - \frac{1}{3 ( 4 + 3 x )} d x)

\int - \frac{1}{3 ( 4 + 3 x )} d x = - \frac{1}{9} ln ∣ 4 + 3 x ∣

= 2 (- \frac{x}{3 ( 4 + 3 x )} - (- \frac{1}{9} ln ∣ 4 + 3 x ∣))

Simplify

= 2 (- \frac{x}{3 ( 4 + 3 x )} + \frac{1}{9} ln ∣ 4 + 3 x ∣)

Add a constant to the solution

= 2 (- \frac{x}{3 ( 4 + 3 x )} + \frac{1}{9} ln ∣ 4 + 3 x ∣) + C

in t e g r a l \frac{( 1 - cos ( x ) ) ^{2}}{sin ^{2} ( x )}

in t e g r a l 3^{- x}

in t e g r a l \frac{arctan ( x )}{x ^{2}}

in t e g r a l \frac{1}{( x - 2 ) ^{3} ( x + 4 ) ^{4}}

in t e g r a l (1 + x tan (x) - sin (x)) sec^{2} (x)

What is the integral of ((2x))/((4+3x)^2) ?
The integral of ((2x))/((4+3x)^2) is 2(-x/(3(4+3x))+1/9 ln|4+3x|)+C