What is the integral of xe^{-x^3} ?

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integral of xe^{-x^3}

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

assuming

v \geq 0

Apply u-substitution

: \int 3 e^{w^{3}} w d w

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{e ^{t^{\frac{3}{2}}}}{2} d t

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

Take the constant out:

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

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

Use the nonelementary integral:

Substitute back

Add a constant to the solution

\int \frac{( 4 x ^{m} - 3 x ^{n} ) ^{3}}{x} d x

x \to \infty lim ((- 1)^{- x})

\int (x^{4} - x^{2} + \frac{1}{4 x ^{3}} - \frac{1}{x ^{2}}) d x

\frac{3}{( s ^{2} + 4 )}

27 n + 3 q^{2} n