What is the integral of (x^{-2}(e^x)^3)/3 ?

The integral of (x^{-2}(e^x)^3)/3 is -(e^{3x})/(3x)+\Ei(3x)+C

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integral of (x^{-2}(e^x)^3)/3

integral

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

- \frac{e ^{3 x}}{3 x} + Ei (3 x) + C

Solution steps

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

Take the constant out:

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

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

Simplify

x^{- 2} (e^{x})^{3} : \frac{e ^{3 x}}{x ^{2}}

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

Apply Integration By Parts

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

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

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

= \frac{1}{3} (- \frac{e ^{3 x}}{x} - (- 3 Ei (3 x)))

Simplify

\frac{1}{3} (- \frac{e ^{3 x}}{x} - (- 3 Ei (3 x))) : - \frac{e ^{3 x}}{3 x} + Ei (3 x)

= - \frac{e ^{3 x}}{3 x} + Ei (3 x)

Add a constant to the solution

= - \frac{e ^{3 x}}{3 x} + Ei (3 x) + C

\frac{1}{( 3 x ^{2} + 1 )}

4 x^{3} + 3 x^{2} + 2

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

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

0.5 ((1 - cos (2 t)))

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