What is the integral of x/((4x+5)^2) ?

The integral of x/((4x+5)^2) is -x/(4(4x+5))+1/16 ln|4x+5|+C

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integral of x/((4x+5)^2)

\int \frac{x}{( 4 x + 5 ) ^{2}} d x

- \frac{x}{4 ( 4 x + 5 )} + \frac{1}{16} ln ∣ 4 x + 5 ∣ + C

Solution steps

\int \frac{x}{( 4 x + 5 ) ^{2}} d x

Apply Integration By Parts

: - \frac{x}{4 ( 4 x + 5 )} - \int - \frac{1}{4 ( 4 x + 5 )} d x

= - \frac{x}{4 ( 4 x + 5 )} - \int - \frac{1}{4 ( 4 x + 5 )} d x

\int - \frac{1}{4 ( 4 x + 5 )} d x = - \frac{1}{16} ln ∣ 4 x + 5 ∣

= - \frac{x}{4 ( 4 x + 5 )} - (- \frac{1}{16} ln ∣ 4 x + 5 ∣)

Simplify

= - \frac{x}{4 ( 4 x + 5 )} + \frac{1}{16} ln ∣ 4 x + 5 ∣

Add a constant to the solution

= - \frac{x}{4 ( 4 x + 5 )} + \frac{1}{16} ln ∣ 4 x + 5 ∣ + C

x \to 1 lim ((x^{3} + 2) (3 - x))

x \to 3 lim (\frac{( 4 x ^{2} + 3 x )}{5})

\frac{10}{2 s + 10}

\int \frac{cos ( \frac{1}{x} )}{x ^{2}} d x

(lo g_{10} (n))^{2}

What is the integral of x/((4x+5)^2) ?
The integral of x/((4x+5)^2) is -x/(4(4x+5))+1/16 ln|4x+5|+C