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

The integral of (sqrt(x)+sqrt(a))^2x is (x^3)/3+4/5 sqrt(a)x^{5/2}+(ax^2)/2+C

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

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

\frac{x ^{3}}{3} + \frac{4}{5} a x^{\frac{5}{2}} + \frac{a x ^{2}}{2} + C

Solution steps

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

Expand

(x + a)^{2} x : x^{2} + 2 a x^{\frac{3}{2}} + a x

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

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

\int 2 a x^{\frac{3}{2}} d x = \frac{4}{5} a x^{\frac{5}{2}}

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

= \frac{x ^{3}}{3} + \frac{4}{5} a x^{\frac{5}{2}} + \frac{a x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{4}{5} a x^{\frac{5}{2}} + \frac{a x ^{2}}{2} + C

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What is the integral of (sqrt(x)+sqrt(a))^2x ?
The integral of (sqrt(x)+sqrt(a))^2x is (x^3)/3+4/5 sqrt(a)x^{5/2}+(ax^2)/2+C