What is the integral of 3x^2e^{-4x} ?

The integral of 3x^2e^{-4x} is -3/32 e^{-4x}(8x^2+4x+1)+C

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integral of 3x^2e^{-4x}

\int 3 x^{2} e^{- 4 x} d x

- \frac{3}{32} e^{- 4 x} (8 x^{2} + 4 x + 1) + C

Solution steps

\int 3 x^{2} e^{- 4 x} d x

Take the constant out:

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

= 3 \cdot \int x^{2} e^{- 4 x} d x

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{64} d u

= 3 \cdot \int - \frac{e ^{u} u ^{2}}{64} d u

Take the constant out:

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

= 3 (- \frac{1}{64} \cdot \int e^{u} u^{2} d u)

Apply Integration By Parts

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

= 3 (- \frac{1}{64} (u^{2} e^{u} - \int 2 u e^{u} d u))

Take the constant out:

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

= 3 (- \frac{1}{64} (u^{2} e^{u} - 2 \cdot \int u e^{u} d u))

Apply Integration By Parts

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

= 3 (- \frac{1}{64} (u^{2} e^{u} - 2 (e^{u} u - \int e^{u} d u)))

Use the common integral:

\int e^{u} d u = e^{u}

= 3 (- \frac{1}{64} (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

Substitute back

u = - 4 x

= 3 (- \frac{1}{64} ((- 4 x)^{2} e^{- 4 x} - 2 (e^{- 4 x} (- 4 x) - e^{- 4 x})))

Simplify

3 (- \frac{1}{64} ((- 4 x)^{2} e^{- 4 x} - 2 (e^{- 4 x} (- 4 x) - e^{- 4 x}))) : - \frac{3}{32} e^{- 4 x} (8 x^{2} + 4 x + 1)

= - \frac{3}{32} e^{- 4 x} (8 x^{2} + 4 x + 1)

Add a constant to the solution

= - \frac{3}{32} e^{- 4 x} (8 x^{2} + 4 x + 1) + C

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What is the integral of 3x^2e^{-4x} ?
The integral of 3x^2e^{-4x} is -3/32 e^{-4x}(8x^2+4x+1)+C