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

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

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

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

a^{2} 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} + 2 a x + x

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

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

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

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

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

Add a constant to the solution

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

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

n = 0 \sum \infty \frac{e ^{n}}{n ^{n}}

\frac{\partial}{\partial y} (sin (x^{2}) - y)

\frac{d}{d x} (2 sin (x))

arcsin (\frac{2}{3})

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