What is the integral of (4x^2)/(sqrt(x)+8) ?

The integral of (4x^2)/(sqrt(x)+8) is

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integral of (4x^2)/(sqrt(x)+8)

\int \frac{4 x ^{2}}{x + 8} d x

8 (\frac{1}{5} (x + 8)^{5} - 10 (x + 8)^{4} + \frac{640}{3} (x + 8)^{3} - 2560 (x + 8)^{2} + 20480 (x + 8) - 32768 ln x + 8) + C

Solution steps

\int \frac{4 x ^{2}}{x + 8} d x

Take the constant out:

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

= 4 \cdot \int \frac{x ^{2}}{x + 8} d x

Apply u-substitution

: \int \frac{2 ( u ^{2} - 16 u + 64 ) ^{2} ( u - 8 )}{u} d u

= 4 \cdot \int \frac{2 ( u ^{2} - 16 u + 64 ) ^{2} ( u - 8 )}{u} d u

Take the constant out:

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

= 4 \cdot 2 \cdot \int \frac{( u ^{2} - 16 u + 64 ) ^{2} ( u - 8 )}{u} d u

Expand

\frac{( u ^{2} - 16 u + 64 ) ^{2} ( u - 8 )}{u} : u^{4} - 40 u^{3} + 640 u^{2} - 5120 u + 20480 - \frac{32768}{u}

= 4 \cdot 2 \cdot \int u^{4} - 40 u^{3} + 640 u^{2} - 5120 u + 20480 - \frac{32768}{u} 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

= 4 \cdot 2 (\int u^{4} d u - \int 40 u^{3} d u + \int 640 u^{2} d u - \int 5120 u d u + \int 20480 d u - \int \frac{32768}{u} d u)

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

\int 40 u^{3} d u = 10 u^{4}

\int 640 u^{2} d u = \frac{640 u ^{3}}{3}

\int 5120 u d u = 2560 u^{2}

\int 20480 d u = 20480 u

\int \frac{32768}{u} d u = 32768 ln ∣ u ∣

= 4 \cdot 2 (\frac{u ^{5}}{5} - 10 u^{4} + \frac{640 u ^{3}}{3} - 2560 u^{2} + 20480 u - 32768 ln ∣ u ∣)

Substitute back

u = x + 8

= 4 \cdot 2 (\frac{( x + 8 ) ^{5}}{5} - 10 (x + 8)^{4} + \frac{640 ( x + 8 ) ^{3}}{3} - 2560 (x + 8)^{2} + 20480 (x + 8) - 32768 ln x + 8)

Simplify

4 \cdot 2 (\frac{( x + 8 ) ^{5}}{5} - 10 (x + 8)^{4} + \frac{640 ( x + 8 ) ^{3}}{3} - 2560 (x + 8)^{2} + 20480 (x + 8) - 32768 ln x + 8) : 8 (\frac{1}{5} (x + 8)^{5} - 10 (x + 8)^{4} + \frac{640}{3} (x + 8)^{3} - 2560 (x + 8)^{2} + 20480 (x + 8) - 32768 ln x + 8)

= 8 (\frac{1}{5} (x + 8)^{5} - 10 (x + 8)^{4} + \frac{640}{3} (x + 8)^{3} - 2560 (x + 8)^{2} + 20480 (x + 8) - 32768 ln x + 8)

Add a constant to the solution

= 8 (\frac{1}{5} (x + 8)^{5} - 10 (x + 8)^{4} + \frac{640}{3} (x + 8)^{3} - 2560 (x + 8)^{2} + 20480 (x + 8) - 32768 ln x + 8) + C

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What is the integral of (4x^2)/(sqrt(x)+8) ?
The integral of (4x^2)/(sqrt(x)+8) is