What is the integral of x^3e^{x^6} ?

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of x^3e^{x^6}

\int x^{3} e^{x^{6}} d x

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

Solution steps

\int x^{3} e^{x^{6}} d x

Apply u-substitution

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

Apply u-substitution

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

Take the constant out:

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

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

Use the nonelementary integral:

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

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

Substitute back

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

Add a constant to the solution

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