What is the integral of xe^{2x^3} ?

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integral of xe^{2x^3}

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

\frac{1}{3 \cdot 2 ^{\frac{2}{3}}} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 2 x ^{3} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 2 x ^{3} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 2 x ^{3} ) ^{2} ) ^{\frac{3}{2}}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the nonelementary integral:

\int e^{x^{a}} d x = - \frac{x Γ ( \frac{1}{a} , - x ^{a} )}{a a - x ^{a}}

= \frac{1}{3 \cdot 2 ^{\frac{2}{3}}} \cdot 3 \cdot \frac{1}{2} - \frac{w Γ ( \frac{1}{\frac{3}{2}} , - w ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - w ^{\frac{3}{2}}}

Substitute back

= \frac{1}{3 \cdot 2 ^{\frac{2}{3}}} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 2 x ^{3} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 2 x ^{3} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 2 x ^{3} ) ^{2} ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{1}{3 \cdot 2 ^{\frac{2}{3}}} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 2 x ^{3} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 2 x ^{3} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 2 x ^{3} ) ^{2} ) ^{\frac{3}{2}}} + C

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