What is the integral of (e^{x^{1/2}})/(x^3) ?

The integral of (e^{x^{1/2}})/(x^3) is

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integral of (e^{x^{1/2}})/(x^3)

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

2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}} + C

Solution steps

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

Apply u-substitution

: \int \frac{2 e ^{u}}{u ^{5}} d u

= \int \frac{2 e ^{u}}{u ^{5}} d u

Take the constant out:

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

= 2 \cdot \int \frac{e ^{u}}{u ^{5}} d u

Apply Integration By Parts

: - \frac{e ^{u}}{4 u ^{4}} - \int - \frac{e ^{u}}{4 u ^{4}} d u

= 2 (- \frac{e ^{u}}{4 u ^{4}} - \int - \frac{e ^{u}}{4 u ^{4}} d u)

\int - \frac{e ^{u}}{4 u ^{4}} d u = - \frac{1}{4} (- \frac{e ^{u}}{3 u ^{3}} + \frac{1}{3} (- \frac{e ^{u}}{2 u ^{2}} + \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))))

= 2 (- \frac{e ^{u}}{4 u ^{4}} - (- \frac{1}{4} (- \frac{e ^{u}}{3 u ^{3}} + \frac{1}{3} (- \frac{e ^{u}}{2 u ^{2}} + \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))))))

Substitute back

u = x^{\frac{1}{2}}

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

Simplify

2 - \frac{e ^{x^{\frac{1}{2}}}}{4 ( x ^{\frac{1}{2}} ) ^{4}} - - \frac{1}{4} - \frac{e ^{x^{\frac{1}{2}}}}{3 ( x ^{\frac{1}{2}} ) ^{3}} + \frac{1}{3} - \frac{e ^{x^{\frac{1}{2}}}}{2 ( x ^{\frac{1}{2}} ) ^{2}} + \frac{1}{2} (- \frac{e ^{x^{\frac{1}{2}}}}{x ^{\frac{1}{2}}} + Ei (x^{\frac{1}{2}})) : 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}}

= 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}}

Add a constant to the solution

= 2 - \frac{e ^{x^{\frac{1}{2}}}}{4 x ^{2}} + \frac{- 2 e ^{x^{\frac{1}{2}}} x + x ^{\frac{3}{2}} ( - e ^{x^{\frac{1}{2}}} + x Ei ( x ^{\frac{1}{2}} ) - e ^{x^{\frac{1}{2}}} x ^{\frac{1}{2}} )}{24 x ^{\frac{5}{2}}} + C

\int (- 2 T + x) \cdot e^{- i \cdot z \cdot x} d x

(t^{2}) \cdot (sin (2 \cdot t))

x \to 25 lim (\frac{\frac{1}{x} - \frac{1}{5}}{x - 25})

\frac{d ^{2} y}{d x ^{2}} - y = 10 sin^{2} (x)

\frac{d}{d x} ((6 x^{3} + 6)^{3} (4 x)^{5 x})

What is the integral of (e^{x^{1/2}})/(x^3) ?
The integral of (e^{x^{1/2}})/(x^3) is