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

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

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integral of 5x^3e^x

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

y = ln (x + 1)

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

\frac{d}{d x} (cos^{- 2} (x))

\frac{d}{d x} ((3 x^{2} + 7 x)^{4})

f (x) = x^{3} - 9 x, at x = 3

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