What is the integral of x^3(e^x)^2 ?

The integral of x^3(e^x)^2 is 1/2 e^{2x}x^3-3/2 (1/2 e^{2x}x^2-1/4 (e^{2x}*2x-e^{2x}))+C

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integral of x^3(e^x)^2

integral

x^{3} (e^{x})^{2}

\frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})) + C

Solution steps

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

(e^{x})^{2} = e^{2 x}

= \int e^{2 x} x^{3} d x

Apply Integration By Parts

: \frac{1}{2} e^{2 x} x^{3} - \int \frac{3}{2} e^{2 x} x^{2} d x

= \frac{1}{2} e^{2 x} x^{3} - \int \frac{3}{2} e^{2 x} x^{2} d x

\int \frac{3}{2} e^{2 x} x^{2} d x = \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

= \frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

Add a constant to the solution

= \frac{1}{2} e^{2 x} x^{3} - \frac{3}{2} (\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})) + C

\frac{1}{( x ^{3} + 1 )}

\frac{( x - 1 )}{( x + 1 ) ^{2}}

- x^{- 2}

\frac{r}{( 1 + r ^{2} ) ^{2}}

sin^{8} (x)

What is the integral of x^3(e^x)^2 ?
The integral of x^3(e^x)^2 is 1/2 e^{2x}x^3-3/2 (1/2 e^{2x}x^2-1/4 (e^{2x}*2x-e^{2x}))+C