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

The integral of (x-4)/(sqrt(x)+2) is (x(x-4))/(sqrt(x)+2)-1/3 x^{3/2}+C

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integral of (x-4)/(sqrt(x)+2)

\int \frac{x - 4}{x + 2} d x

\frac{x ( x - 4 )}{x + 2} - \frac{1}{3} x^{\frac{3}{2}} + C

Solution steps

\int \frac{x - 4}{x + 2} d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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