What is the integral of (x^2)/(sqrt(x+3)) ?

The integral of (x^2)/(sqrt(x+3)) is 2(1/5 (x+3)^{5/2}-2(x+3)^{3/2}+9sqrt(x+3))+C

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integral of (x^2)/(sqrt(x+3))

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

2 (\frac{1}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + 9 x + 3) + C

Solution steps

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

Apply u-substitution

: \int 2 (u^{2} - 3)^{2} d u

= \int 2 (u^{2} - 3)^{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int (u^{2} - 3)^{2} d u

Expand

(u^{2} - 3)^{2} : u^{4} - 6 u^{2} + 9

= 2 \cdot \int u^{4} - 6 u^{2} + 9 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int u^{4} d u - \int 6 u^{2} d u + \int 9 d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 6 u^{2} d u = 2 u^{3}

\int 9 d u = 9 u

= 2 (\frac{u ^{5}}{5} - 2 u^{3} + 9 u)

Substitute back

u = x + 3

= 2 (\frac{( x + 3 ) ^{5}}{5} - 2 (x + 3)^{3} + 9 x + 3)

Simplify

2 (\frac{( x + 3 ) ^{5}}{5} - 2 (x + 3)^{3} + 9 x + 3) : 2 (\frac{1}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + 9 x + 3)

= 2 (\frac{1}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + 9 x + 3)

Add a constant to the solution

= 2 (\frac{1}{5} (x + 3)^{\frac{5}{2}} - 2 (x + 3)^{\frac{3}{2}} + 9 x + 3) + C

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

y = 11 x, y = x^{2} - 12

\int_{1}^{4} (x^{2}) d x

\int_{1}^{3} 1 + \frac{1}{x ^{2}} d x

n = 0 \sum \infty 0. 4^{n}

What is the integral of (x^2)/(sqrt(x+3)) ?
The integral of (x^2)/(sqrt(x+3)) is 2(1/5 (x+3)^{5/2}-2(x+3)^{3/2}+9sqrt(x+3))+C