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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

= - \frac{e ^{3 x}}{2 x ^{2}} + \frac{3}{2} (- \frac{e ^{3 x}}{x} + 3 Ei (3 x)) + C

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