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

The integral of x^5e^x is 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 x^5e^x

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

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 x^{5} e^{x} d x

Apply Integration By Parts

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

= 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}))))

= 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

= 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

\frac{d}{d x} (1 x^{2})

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

\frac{d}{d x} ((x - 3)^{2})

\int_{1}^{2} sin (x) d x

\frac{d}{d x} ((x^{2} - 4)^{2})

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