What is the integral of (sqrt(a)-sqrt(x))^2 ?

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

integral of (sqrt(a)-sqrt(x))^2

\int (a - x)^{2} d x

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

Solution steps

\int (a - x)^{2} d x

Expand

(a - x)^{2} : a - 2 a x + x

= \int a - 2 a x + x 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 a d x - \int 2 a x d x + \int x d x

\int a d x = a x

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

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

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

Add a constant to the solution

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

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