What is the integral of x^3e^{9x} ?

The integral of x^3e^{9x} is 1/2187 e^{9x}(243x^3-81x^2+18x-2)+C

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integral of x^3e^{9x}

\int x^{3} e^{9 x} d x

\frac{1}{2187} e^{9 x} (243 x^{3} - 81 x^{2} + 18 x - 2) + C

Solution steps

\int x^{3} e^{9 x} d x

Apply u-substitution

: \int \frac{e ^{u} u ^{3}}{6561} d u

= \int \frac{e ^{u} u ^{3}}{6561} d u

Take the constant out:

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

= \frac{1}{6561} \cdot \int e^{u} u^{3} d u

Apply Integration By Parts

: u^{3} e^{u} - \int 3 u^{2} e^{u} d u

= \frac{1}{6561} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= \frac{1}{6561} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = 9 x

= \frac{1}{6561} ((9 x)^{3} e^{9 x} - 3 ((9 x)^{2} e^{9 x} - 2 (e^{9 x} \cdot 9 x - e^{9 x})))

Simplify

\frac{1}{6561} ((9 x)^{3} e^{9 x} - 3 ((9 x)^{2} e^{9 x} - 2 (e^{9 x} \cdot 9 x - e^{9 x}))) : \frac{1}{2187} e^{9 x} (243 x^{3} - 81 x^{2} + 18 x - 2)

= \frac{1}{2187} e^{9 x} (243 x^{3} - 81 x^{2} + 18 x - 2)

Add a constant to the solution

= \frac{1}{2187} e^{9 x} (243 x^{3} - 81 x^{2} + 18 x - 2) + C

y = 10 x^{4} - 10 x^{2}, y = 7 x^{2}

x + y \cdot \frac{d y}{d x} = 0

(0.3) (4.1)

y = e^{x}, y = e^{- 4 x}, x = ln (4)

\int \frac{1}{x ( x + 1 ) ( x + 2 )} d x

What is the integral of x^3e^{9x} ?
The integral of x^3e^{9x} is 1/2187 e^{9x}(243x^3-81x^2+18x-2)+C