What is the integral of x^23^{x^3} ?

The integral of x^23^{x^3} is (3^{x^3-1})/(ln(3))+C

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integral of x^23^{x^3}

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

\frac{3 ^{x^{3} - 1}}{ln ( 3 )} + C

Solution steps

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

Apply u-substitution

: \int \frac{u}{ln ( 3 ) \cdot 3 ^{\frac{l n ( u ) + l n ( 3 )}{l n ( 3 )}}} d u

= \int \frac{u}{ln ( 3 ) \cdot 3 ^{\frac{l n ( u ) + l n ( 3 )}{l n ( 3 )}}} d u

Take the constant out:

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

= \frac{1}{ln ( 3 )} \cdot \int \frac{u}{3 ^{\frac{l n ( u ) + l n ( 3 )}{l n ( 3 )}}} d u

Apply Integration By Parts

: \frac{u ^{2}}{2 \cdot 3 ^{\frac{l n ( u ) + l n ( 3 )}{l n ( 3 )}}} - \int - \frac{1}{6} d u

= \frac{1}{ln ( 3 )} (\frac{u ^{2}}{2 \cdot 3 ^{\frac{l n ( u ) + l n ( 3 )}{l n ( 3 )}}} - \int - \frac{1}{6} d u)

\int - \frac{1}{6} d u = - \frac{1}{6} u

= \frac{1}{ln ( 3 )} (\frac{u ^{2}}{2 \cdot 3 ^{\frac{l n ( u ) + l n ( 3 )}{l n ( 3 )}}} - (- \frac{1}{6} u))

Substitute back

u = 3^{x^{3}}

= \frac{1}{ln ( 3 )} \frac{( 3 ^{x^{3}} ) ^{2}}{2 \cdot 3 ^{\frac{l n ( 3 ^{x^{3}} ) + l n ( 3 )}{l n ( 3 )}}} - (- \frac{1}{6} \cdot 3^{x^{3}})

Simplify

\frac{1}{ln ( 3 )} \frac{( 3 ^{x^{3}} ) ^{2}}{2 \cdot 3 ^{\frac{l n ( 3 ^{x^{3}} ) + l n ( 3 )}{l n ( 3 )}}} - (- \frac{1}{6} \cdot 3^{x^{3}}) : \frac{3 ^{x^{3} - 1}}{ln ( 3 )}

= \frac{3 ^{x^{3} - 1}}{ln ( 3 )}

Add a constant to the solution

= \frac{3 ^{x^{3} - 1}}{ln ( 3 )} + C

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What is the integral of x^23^{x^3} ?
The integral of x^23^{x^3} is (3^{x^3-1})/(ln(3))+C