What is the integral of 3x^5e^x ?

The integral of 3x^5e^x is 3(x^5e^x-5(x^4e^x-4(x^3e^x-3(x^2e^x-2(e^xx-e^x)))))+C

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integral of 3x^5e^x

\int 3 x^{5} e^{x} d x

3 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))) + C

Solution steps

\int 3 x^{5} e^{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^{5} e^{x} d x

Apply Integration By Parts

: x^{5} e^{x} - \int 5 x^{4} e^{x} d x

= 3 (x^{5} e^{x} - \int 5 x^{4} e^{x} d x)

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

= 3 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x})))))

Add a constant to the solution

= 3 (x^{5} e^{x} - 5 (x^{4} e^{x} - 4 (x^{3} e^{x} - 3 (x^{2} e^{x} - 2 (e^{x} x - e^{x}))))) + C

6 x + \frac{d y}{d x} = 0

\frac{\partial}{\partial x} (x (x^{2} + y^{2} + z^{2})^{m})

\frac{d}{d x} (\frac{1}{f ( x )})

\int_{1}^{- 1} (x - 1) (3 x + 2) d x

What is the integral of 3x^5e^x ?
The integral of 3x^5e^x is 3(x^5e^x-5(x^4e^x-4(x^3e^x-3(x^2e^x-2(e^xx-e^x)))))+C