What is the integral of x^2e^{-12x} ?

The integral of x^2e^{-12x} is -1/864 e^{-12x}(72x^2+12x+1)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^2e^{-12x}

\int x^{2} e^{- 12 x} d x

- \frac{1}{864} e^{- 12 x} (72 x^{2} + 12 x + 1) + C

Solution steps

\int x^{2} e^{- 12 x} d x

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{1728} d u

= \int - \frac{e ^{u} u ^{2}}{1728} d u

Take the constant out:

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

= - \frac{1}{1728} \cdot \int e^{u} u^{2} d u

Apply Integration By Parts

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

= - \frac{1}{1728} (u^{2} e^{u} - \int 2 u e^{u} d u)

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

= - \frac{1}{1728} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 12 x

= - \frac{1}{1728} ((- 12 x)^{2} e^{- 12 x} - 2 (e^{- 12 x} (- 12 x) - e^{- 12 x}))

Simplify

- \frac{1}{1728} ((- 12 x)^{2} e^{- 12 x} - 2 (e^{- 12 x} (- 12 x) - e^{- 12 x})) : - \frac{1}{864} e^{- 12 x} (72 x^{2} + 12 x + 1)

= - \frac{1}{864} e^{- 12 x} (72 x^{2} + 12 x + 1)

Add a constant to the solution

= - \frac{1}{864} e^{- 12 x} (72 x^{2} + 12 x + 1) + C

\int \frac{1}{x ^{2} + 8 x + 3} d x

\int \frac{1}{x ( ln ( x ) ) ^{\frac{3}{2}}} d x

\int (e^{3 x - 4}) d x

\int (4 + x^{2})^{\frac{3}{2}} d x

\int \frac{e ^{- x} cosh ( x )}{2} d x

What is the integral of x^2e^{-12x} ?
The integral of x^2e^{-12x} is -1/864 e^{-12x}(72x^2+12x+1)+C