What is the integral of (x^{1/2}-2)^2 ?

The integral of (x^{1/2}-2)^2 is (x^2)/2-8/3 x^{3/2}+4x+C

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integral of (x^{1/2}-2)^2

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

\frac{x ^{2}}{2} - \frac{8}{3} x^{\frac{3}{2}} + 4 x + C

Solution steps

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

Expand

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

= \int x - 4 x^{\frac{1}{2}} + 4 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 x d x - \int 4 x^{\frac{1}{2}} d x + \int 4 d x

\int x d x = \frac{x ^{2}}{2}

\int 4 x^{\frac{1}{2}} d x = \frac{8}{3} x^{\frac{3}{2}}

\int 4 d x = 4 x

= \frac{x ^{2}}{2} - \frac{8}{3} x^{\frac{3}{2}} + 4 x

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{8}{3} x^{\frac{3}{2}} + 4 x + C

What is the integral of (x^{1/2}-2)^2 ?
The integral of (x^{1/2}-2)^2 is (x^2)/2-8/3 x^{3/2}+4x+C