What is the integral of 5x^2sqrt(x+9) ?

The integral of 5x^2sqrt(x+9) is 10/7 (x+9)^{7/2}-36(x+9)^{5/2}+270(x+9)^{3/2}+C

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integral of 5x^2sqrt(x+9)

\int 5 x^{2} x + 9 d x

\frac{10}{7} (x + 9)^{\frac{7}{2}} - 36 (x + 9)^{\frac{5}{2}} + 270 (x + 9)^{\frac{3}{2}} + C

Solution steps

\int 5 x^{2} x + 9 d x

Take the constant out:

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

= 5 \cdot \int x^{2} x + 9 d x

Apply u-substitution

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

= 5 \cdot \int (u - 9)^{2} u d u

Expand

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

= 5 \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

= 5 (\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}}

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

Substitute back

u = x + 9

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

Simplify

5 (\frac{2}{7} (x + 9)^{\frac{7}{2}} - \frac{36}{5} (x + 9)^{\frac{5}{2}} + 54 (x + 9)^{\frac{3}{2}}) : \frac{10}{7} (x + 9)^{\frac{7}{2}} - 36 (x + 9)^{\frac{5}{2}} + 270 (x + 9)^{\frac{3}{2}}

= \frac{10}{7} (x + 9)^{\frac{7}{2}} - 36 (x + 9)^{\frac{5}{2}} + 270 (x + 9)^{\frac{3}{2}}

Add a constant to the solution

= \frac{10}{7} (x + 9)^{\frac{7}{2}} - 36 (x + 9)^{\frac{5}{2}} + 270 (x + 9)^{\frac{3}{2}} + C

f (x) = 9 x + 4 x^{- 1}

\int_{0}^{1} (y^{6} + y^{8}) d y

x \to 1 + lim (\frac{1}{x} - 1)

y = 4 x, y = 6 x^{2}

What is the integral of 5x^2sqrt(x+9) ?
The integral of 5x^2sqrt(x+9) is 10/7 (x+9)^{7/2}-36(x+9)^{5/2}+270(x+9)^{3/2}+C