What is the integral of (5e^{x^{-2}})/(x^3) ?

The integral of (5e^{x^{-2}})/(x^3) is -5/2 e^{1/(x^2)}+C

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integral of (5e^{x^{-2}})/(x^3)

\int \frac{5 e ^{x^{- 2}}}{x ^{3}} d x

- \frac{5}{2} e^{\frac{1}{x ^{2}}} + C

Solution steps

\int \frac{5 e ^{x^{- 2}}}{x ^{3}} d x

Take the constant out:

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

= 5 \cdot \int \frac{e ^{x^{- 2}}}{x ^{3}} d x

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

= 5 \cdot \int \frac{e ^{\frac{1}{x ^{2}}}}{x ^{3}} d x

Apply u-substitution

: \int \frac{e ^{\frac{1}{u}}}{2 u ^{2}} d u

= 5 \cdot \int \frac{e ^{\frac{1}{u}}}{2 u ^{2}} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{e ^{\frac{1}{u}}}{u ^{2}} d u

Apply u-substitution

: \int - e^{v} d v

= 5 \cdot \frac{1}{2} \cdot \int - e^{v} d v

Take the constant out:

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

= 5 \cdot \frac{1}{2} (- \int e^{v} d v)

Use the common integral:

\int e^{v} d v = e^{v}

= 5 \cdot \frac{1}{2} (- e^{v})

Substitute back

= 5 \cdot \frac{1}{2} (- e^{\frac{1}{x ^{2}}})

Simplify

5 \cdot \frac{1}{2} (- e^{\frac{1}{x ^{2}}}) : - \frac{5}{2} e^{\frac{1}{x ^{2}}}

= - \frac{5}{2} e^{\frac{1}{x ^{2}}}

Add a constant to the solution

= - \frac{5}{2} e^{\frac{1}{x ^{2}}} + C

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What is the integral of (5e^{x^{-2}})/(x^3) ?
The integral of (5e^{x^{-2}})/(x^3) is -5/2 e^{1/(x^2)}+C