What is the integral of x^6e^{-4x} ?

The integral of x^6e^{-4x} is -1/1024 e^{-4x}(256x^6+3(128x^5+5(32x^4+32x^3+3(8x^2+4x+1))))+C

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integral of x^6e^{-4x}

\int x^{6} e^{- 4 x} d x

- \frac{1}{1024} e^{- 4 x} (256 x^{6} + 3 (128 x^{5} + 5 (32 x^{4} + 32 x^{3} + 3 (8 x^{2} + 4 x + 1)))) + C

Solution steps

\int x^{6} e^{- 4 x} d x

Apply u-substitution

: \int - \frac{e ^{u} u ^{6}}{16384} d u

= \int - \frac{e ^{u} u ^{6}}{16384} d u

Take the constant out:

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

= - \frac{1}{16384} \cdot \int e^{u} u^{6} d u

Apply Integration By Parts

: u^{6} e^{u} - \int 6 u^{5} e^{u} d u

= - \frac{1}{16384} (u^{6} e^{u} - \int 6 u^{5} e^{u} d u)

\int 6 u^{5} e^{u} d u = 6 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

= - \frac{1}{16384} (u^{6} e^{u} - 6 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))))))

Substitute back

u = - 4 x

= - \frac{1}{16384} ((- 4 x)^{6} e^{- 4 x} - 6 ((- 4 x)^{5} e^{- 4 x} - 5 ((- 4 x)^{4} e^{- 4 x} - 4 ((- 4 x)^{3} e^{- 4 x} - 3 ((- 4 x)^{2} e^{- 4 x} - 2 (e^{- 4 x} (- 4 x) - e^{- 4 x}))))))

Simplify

- \frac{1}{16384} ((- 4 x)^{6} e^{- 4 x} - 6 ((- 4 x)^{5} e^{- 4 x} - 5 ((- 4 x)^{4} e^{- 4 x} - 4 ((- 4 x)^{3} e^{- 4 x} - 3 ((- 4 x)^{2} e^{- 4 x} - 2 (e^{- 4 x} (- 4 x) - e^{- 4 x})))))) : - \frac{1}{1024} e^{- 4 x} (256 x^{6} + 3 (128 x^{5} + 5 (32 x^{4} + 32 x^{3} + 3 (8 x^{2} + 4 x + 1))))

= - \frac{1}{1024} e^{- 4 x} (256 x^{6} + 3 (128 x^{5} + 5 (32 x^{4} + 32 x^{3} + 3 (8 x^{2} + 4 x + 1))))

Add a constant to the solution

= - \frac{1}{1024} e^{- 4 x} (256 x^{6} + 3 (128 x^{5} + 5 (32 x^{4} + 32 x^{3} + 3 (8 x^{2} + 4 x + 1)))) + C

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e^{- cs}

What is the integral of x^6e^{-4x} ?
The integral of x^6e^{-4x} is -1/1024 e^{-4x}(256x^6+3(128x^5+5(32x^4+32x^3+3(8x^2+4x+1))))+C