What is the integral of x^2sqrt(5x+6) ?

The integral of x^2sqrt(5x+6) is 1/125 (2/7 (5x+6)^{7/2}-24/5 (5x+6)^{5/2}+24(5x+6)^{3/2})+C

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integral of x^2sqrt(5x+6)

\int x^{2} 5 x + 6 d x

\frac{1}{125} (\frac{2}{7} (5 x + 6)^{\frac{7}{2}} - \frac{24}{5} (5 x + 6)^{\frac{5}{2}} + 24 (5 x + 6)^{\frac{3}{2}}) + C

Solution steps

\int x^{2} 5 x + 6 d x

Apply u-substitution

: \int \frac{u ( u - 6 ) ^{2}}{125} d u

= \int \frac{u ( u - 6 ) ^{2}}{125} d u

Take the constant out:

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

= \frac{1}{125} \cdot \int u (u - 6)^{2} d u

Expand

u (u - 6)^{2} : u^{\frac{5}{2}} - 12 u^{\frac{3}{2}} + 36 u

= \frac{1}{125} \cdot \int u^{\frac{5}{2}} - 12 u^{\frac{3}{2}} + 36 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

= \frac{1}{125} (\int u^{\frac{5}{2}} d u - \int 12 u^{\frac{3}{2}} d u + \int 36 u d u)

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

\int 12 u^{\frac{3}{2}} d u = \frac{24}{5} u^{\frac{5}{2}}

\int 36 u d u = 24 u^{\frac{3}{2}}

= \frac{1}{125} (\frac{2}{7} u^{\frac{7}{2}} - \frac{24}{5} u^{\frac{5}{2}} + 24 u^{\frac{3}{2}})

Substitute back

u = 5 x + 6

= \frac{1}{125} (\frac{2}{7} (5 x + 6)^{\frac{7}{2}} - \frac{24}{5} (5 x + 6)^{\frac{5}{2}} + 24 (5 x + 6)^{\frac{3}{2}})

Add a constant to the solution

= \frac{1}{125} (\frac{2}{7} (5 x + 6)^{\frac{7}{2}} - \frac{24}{5} (5 x + 6)^{\frac{5}{2}} + 24 (5 x + 6)^{\frac{3}{2}}) + C

x \to 0 lim (C e^{- 4 x})

x \to - 1 lim (\frac{x ^{2} + 1}{x})

\frac{d}{d x} (\frac{x y}{x ^{2} + y ^{2}})

\int t^{3} e^{t^{4}} d t

y^{^{''}} + 16 y = 2 sec (4 t)

What is the integral of x^2sqrt(5x+6) ?
The integral of x^2sqrt(5x+6) is 1/125 (2/7 (5x+6)^{7/2}-24/5 (5x+6)^{5/2}+24(5x+6)^{3/2})+C