What is the integral of x^2e^{-7x} ?

The integral of x^2e^{-7x} is -1/343 (49e^{-7x}x^2-2(-7e^{-7x}x-e^{-7x}))+C

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integral of x^2e^{-7x}

\int x^{2} e^{- 7 x} d x

- \frac{1}{343} (49 e^{- 7 x} x^{2} - 2 (- 7 e^{- 7 x} x - e^{- 7 x})) + C

Solution steps

\int x^{2} e^{- 7 x} d x

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{343} d u

= \int - \frac{e ^{u} u ^{2}}{343} d u

Take the constant out:

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

= - \frac{1}{343} \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}{343} (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}{343} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 7 x

= - \frac{1}{343} ((- 7 x)^{2} e^{- 7 x} - 2 (e^{- 7 x} (- 7 x) - e^{- 7 x}))

Simplify

- \frac{1}{343} ((- 7 x)^{2} e^{- 7 x} - 2 (e^{- 7 x} (- 7 x) - e^{- 7 x})) : - \frac{1}{343} (49 e^{- 7 x} x^{2} - 2 (- 7 e^{- 7 x} x - e^{- 7 x}))

= - \frac{1}{343} (49 e^{- 7 x} x^{2} - 2 (- 7 e^{- 7 x} x - e^{- 7 x}))

Add a constant to the solution

= - \frac{1}{343} (49 e^{- 7 x} x^{2} - 2 (- 7 e^{- 7 x} x - e^{- 7 x})) + C

\frac{\partial}{\partial s} (r sin (s))

2 y^{^{''}} + y^{^{'}} - y = 0

2 e^{x^{3}} + 1

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

\frac{\partial}{\partial y} (y^{2} - cos (x y))

What is the integral of x^2e^{-7x} ?
The integral of x^2e^{-7x} is -1/343 (49e^{-7x}x^2-2(-7e^{-7x}x-e^{-7x}))+C