What is the integral of 4x^5e^x ?

The integral of 4x^5e^x is 4(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 4x^5e^x

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

4 (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 4 x^{5} 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^{5} e^{x} d x

Apply Integration By Parts

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

= 4 (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}))))

= 4 (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

= 4 (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

\int \frac{x ^{4} + x}{x} d x

cos (x^{2})

\int \frac{sin ( \frac{1}{x ^{5}} )}{x ^{6}} d x

\int x 2 + x^{2} d x

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

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