What is the integral of x^3e^{8x} ?

The integral of x^3e^{8x} is 1/4096 (512e^{8x}x^3-3(64e^{8x}x^2-2(8e^{8x}x-e^{8x})))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^3e^{8x}

\int x^{3} e^{8 x} d x

\frac{1}{4096} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x}))) + C

Solution steps

\int x^{3} e^{8 x} d x

Apply u-substitution

: \int \frac{e ^{u} u ^{3}}{4096} d u

= \int \frac{e ^{u} u ^{3}}{4096} d u

Take the constant out:

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

= \frac{1}{4096} \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}{4096} (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}{4096} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = 8 x

= \frac{1}{4096} ((8 x)^{3} e^{8 x} - 3 ((8 x)^{2} e^{8 x} - 2 (e^{8 x} \cdot 8 x - e^{8 x})))

Simplify

\frac{1}{4096} ((8 x)^{3} e^{8 x} - 3 ((8 x)^{2} e^{8 x} - 2 (e^{8 x} \cdot 8 x - e^{8 x}))) : \frac{1}{4096} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))

= \frac{1}{4096} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x})))

Add a constant to the solution

= \frac{1}{4096} (512 e^{8 x} x^{3} - 3 (64 e^{8 x} x^{2} - 2 (8 e^{8 x} x - e^{8 x}))) + C

y^{^{''}} - 14 y^{^{'}} + 49 y = 0

y^{^{'''}} + 4 y^{^{'}} = 0

(2.5) (4.7)

\frac{d}{d x} (9 e^{4 x})

\frac{\partial}{\partial x} (\frac{2 x ^{2} + 7 x ^{2} y}{3 x y})

What is the integral of x^3e^{8x} ?
The integral of x^3e^{8x} is 1/4096 (512e^{8x}x^3-3(64e^{8x}x^2-2(8e^{8x}x-e^{8x})))+C