What is the integral of (sqrt(x))/((4x+9)^2) ?

The integral of (sqrt(x))/((4x+9)^2) is -(sqrt(x))/(4(4x+9))+1/24 arctan(2/3 sqrt(x))+C

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integral of (sqrt(x))/((4x+9)^2)

\int \frac{x}{( 4 x + 9 ) ^{2}} d x

- \frac{x}{4 ( 4 x + 9 )} + \frac{1}{24} arctan (\frac{2}{3} x) + C

Solution steps

\int \frac{x}{( 4 x + 9 ) ^{2}} d x

Apply Integration By Parts

: - \frac{x}{4 ( 4 x + 9 )} - \int - \frac{1}{8 x ( 4 x + 9 )} d x

= - \frac{x}{4 ( 4 x + 9 )} - \int - \frac{1}{8 x ( 4 x + 9 )} d x

\int - \frac{1}{8 x ( 4 x + 9 )} d x = - \frac{1}{24} arctan (\frac{2}{3} x)

= - \frac{x}{4 ( 4 x + 9 )} - (- \frac{1}{24} arctan (\frac{2}{3} x))

Simplify

= - \frac{x}{4 ( 4 x + 9 )} + \frac{1}{24} arctan (\frac{2}{3} x)

Add a constant to the solution

= - \frac{x}{4 ( 4 x + 9 )} + \frac{1}{24} arctan (\frac{2}{3} x) + C

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What is the integral of (sqrt(x))/((4x+9)^2) ?
The integral of (sqrt(x))/((4x+9)^2) is -(sqrt(x))/(4(4x+9))+1/24 arctan(2/3 sqrt(x))+C