What is the integral of x^3*6^{x^4} ?

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

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integral of x^3*6^{x^4}

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

\frac{3 ^{x^{4}} \cdot 2 ^{x^{4} - 2}}{ln ( 6 )} + C

Solution steps

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

Apply u-substitution

: \int 2^{u^{\frac{4}{3}}} \cdot 3^{u^{\frac{4}{3}} - 1} u^{\frac{1}{3}} d u

= \int 2^{u^{\frac{4}{3}}} \cdot 3^{u^{\frac{4}{3}} - 1} u^{\frac{1}{3}} d u

Apply u-substitution

: \int 3^{v} \cdot 2^{v - 2} d v

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

Simplify

2^{v - 2} : 2^{v} \cdot 2^{- 2}

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

Take the constant out:

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

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

2^{- 2} = \frac{1}{4}

= \frac{1}{4} \cdot \int 3^{v} \cdot 2^{v} d v

Simplify

3^{v} \cdot 2^{v} : 6^{v}

= \frac{1}{4} \cdot \int 6^{v} d v

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{4} \cdot \frac{6 ^{v}}{ln ( 6 )}

Substitute back

= \frac{1}{4} \cdot \frac{6 ^{(x^{3})^{\frac{4}{3}}}}{ln ( 6 )}

Simplify

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

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

Add a constant to the solution

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

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