What is the integral of (x^2)/(sqrt(x+5)) ?

The integral of (x^2)/(sqrt(x+5)) is 2(1/5 (x+5)^{5/2}-10/3 (x+5)^{3/2}+25sqrt(x+5))+C

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integral of (x^2)/(sqrt(x+5))

\int \frac{x ^{2}}{x + 5} d x

2 (\frac{1}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}} + 25 x + 5) + C

Solution steps

\int \frac{x ^{2}}{x + 5} d x

Apply u-substitution

: \int 2 (u^{2} - 5)^{2} d u

= \int 2 (u^{2} - 5)^{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} - 5)^{2} d u

Expand

(u^{2} - 5)^{2} : u^{4} - 10 u^{2} + 25

= 2 \cdot \int u^{4} - 10 u^{2} + 25 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 10 u^{2} d u + \int 25 d u)

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

\int 10 u^{2} d u = \frac{10 u ^{3}}{3}

\int 25 d u = 25 u

= 2 (\frac{u ^{5}}{5} - \frac{10 u ^{3}}{3} + 25 u)

Substitute back

u = x + 5

= 2 (\frac{( x + 5 ) ^{5}}{5} - \frac{10 ( x + 5 ) ^{3}}{3} + 25 x + 5)

Simplify

2 (\frac{( x + 5 ) ^{5}}{5} - \frac{10 ( x + 5 ) ^{3}}{3} + 25 x + 5) : 2 (\frac{1}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}} + 25 x + 5)

= 2 (\frac{1}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}} + 25 x + 5)

Add a constant to the solution

= 2 (\frac{1}{5} (x + 5)^{\frac{5}{2}} - \frac{10}{3} (x + 5)^{\frac{3}{2}} + 25 x + 5) + C

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What is the integral of (x^2)/(sqrt(x+5)) ?
The integral of (x^2)/(sqrt(x+5)) is 2(1/5 (x+5)^{5/2}-10/3 (x+5)^{3/2}+25sqrt(x+5))+C