What is the integral of 4x^3e^x ?

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

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integral of 4x^3e^x

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

= 4 (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}))

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

Add a constant to the solution

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

\frac{1}{x}, 0, y = 1, y = e

x \to 3 lim (x^{3} + 6 x)

\int \frac{x - 4}{x ^{2} + 5 x + 6} d x

y = (\frac{cos ( x )}{1 - sin ( x )})^{2}

n = 1 \sum \infty \frac{1}{3 ^{n} + n ^{3}}

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