What is the integral of 6x^24^{x^3} ?

The integral of 6x^24^{x^3} is (2^{4x^3+2})/(ln(2)*4^{x^3+1)}+C

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integral of 6x^24^{x^3}

\int 6 x^{2} 4^{x^{3}} d x

\frac{2 ^{4 x^{3} + 2}}{ln ( 2 ) \cdot 4 ^{x^{3} + 1}} + C

Solution steps

\int 6 x^{2} \cdot 4^{x^{3}} d x

Take the constant out:

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

= 6 \cdot \int x^{2} \cdot 4^{x^{3}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} - \int - \frac{1}{4} d u

= 6 \cdot \frac{1}{3 ln ( 2 )} (\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} - \int - \frac{1}{4} d u)

\int - \frac{1}{4} d u = - \frac{1}{4} u

= 6 \cdot \frac{1}{3 ln ( 2 )} (\frac{u ^{2}}{2 ^{\frac{l n ( 2 u ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} u))

Substitute back

u = 4^{x^{3}}

= 6 \cdot \frac{1}{3 ln ( 2 )} \frac{( 4 ^{x^{3}} ) ^{2}}{2 ^{\frac{l n ( 2 \cdot 4 ^{x^{3}} ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} \cdot 4^{x^{3}})

Simplify

6 \cdot \frac{1}{3 ln ( 2 )} \frac{( 4 ^{x^{3}} ) ^{2}}{2 ^{\frac{l n ( 2 \cdot 4 ^{x^{3}} ) + l n ( 2 )}{l n ( 2 )}}} - (- \frac{1}{4} \cdot 4^{x^{3}}) : \frac{2 ^{4 x^{3} + 2}}{ln ( 2 ) \cdot 4 ^{x^{3} + 1}}

= \frac{2 ^{4 x^{3} + 2}}{ln ( 2 ) \cdot 4 ^{x^{3} + 1}}

Add a constant to the solution

= \frac{2 ^{4 x^{3} + 2}}{ln ( 2 ) \cdot 4 ^{x^{3} + 1}} + C

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What is the integral of 6x^24^{x^3} ?
The integral of 6x^24^{x^3} is (2^{4x^3+2})/(ln(2)*4^{x^3+1)}+C