What is the integral of 4x^2sqrt(x+9) ?

The integral of 4x^2sqrt(x+9) is 4(2/7 (x+9)^{7/2}-36/5 (x+9)^{5/2}+54(x+9)^{3/2})+C

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integral of 4x^2sqrt(x+9)

\int 4 x^{2} x + 9 d x

4 (\frac{2}{7} (x + 9)^{\frac{7}{2}} - \frac{36}{5} (x + 9)^{\frac{5}{2}} + 54 (x + 9)^{\frac{3}{2}}) + C

Solution steps

\int 4 x^{2} x + 9 d x

Take the constant out:

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

= 4 \cdot \int x^{2} x + 9 d x

Apply u-substitution

: \int (u - 9)^{2} u d u

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

Expand

(u - 9)^{2} u : u^{\frac{5}{2}} - 18 u^{\frac{3}{2}} + 81 u

= 4 \cdot \int u^{\frac{5}{2}} - 18 u^{\frac{3}{2}} + 81 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 (\int u^{\frac{5}{2}} d u - \int 18 u^{\frac{3}{2}} d u + \int 81 u d u)

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int 18 u^{\frac{3}{2}} d u = \frac{36}{5} u^{\frac{5}{2}}

\int 81 u d u = 54 u^{\frac{3}{2}}

= 4 (\frac{2}{7} u^{\frac{7}{2}} - \frac{36}{5} u^{\frac{5}{2}} + 54 u^{\frac{3}{2}})

Substitute back

u = x + 9

= 4 (\frac{2}{7} (x + 9)^{\frac{7}{2}} - \frac{36}{5} (x + 9)^{\frac{5}{2}} + 54 (x + 9)^{\frac{3}{2}})

Add a constant to the solution

= 4 (\frac{2}{7} (x + 9)^{\frac{7}{2}} - \frac{36}{5} (x + 9)^{\frac{5}{2}} + 54 (x + 9)^{\frac{3}{2}}) + C

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What is the integral of 4x^2sqrt(x+9) ?
The integral of 4x^2sqrt(x+9) is 4(2/7 (x+9)^{7/2}-36/5 (x+9)^{5/2}+54(x+9)^{3/2})+C