What is the integral of sqrt(x^47^{4x^3)} ?

The integral of sqrt(x^47^{4x^3)} is (49^{x^3})/(6ln(7))+C

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integral of sqrt(x^47^{4x^3)}

\int x^{4} 7^{4 x^{3}} d x

\frac{4 9 ^{x^{3}}}{6 ln ( 7 )} + C

Solution steps

\int x^{4} \cdot 7^{4 x^{3}} d x

Apply u-substitution

: \int \frac{7 ^{\frac{u}{2}}}{12} d u

= \int \frac{7 ^{\frac{u}{2}}}{12} d u

Take the constant out:

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

= \frac{1}{12} \cdot \int 7^{\frac{u}{2}} d u

Apply u-substitution

: \int 7^{v} \cdot 2 d v

= \frac{1}{12} \cdot \int 7^{v} \cdot 2 d v

Take the constant out:

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

= \frac{1}{12} \cdot 2 \cdot \int 7^{v} d v

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

= \frac{1}{12} \cdot 2 \cdot \frac{7 ^{v}}{ln ( 7 )}

Substitute back

= \frac{1}{12} \cdot 2 \cdot \frac{7 ^{\frac{4 x ^{3}}{2}}}{ln ( 7 )}

Simplify

\frac{1}{12} \cdot 2 \cdot \frac{7 ^{\frac{4 x ^{3}}{2}}}{ln ( 7 )} : \frac{4 9 ^{x^{3}}}{6 ln ( 7 )}

= \frac{4 9 ^{x^{3}}}{6 ln ( 7 )}

Add a constant to the solution

= \frac{4 9 ^{x^{3}}}{6 ln ( 7 )} + C

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What is the integral of sqrt(x^47^{4x^3)} ?
The integral of sqrt(x^47^{4x^3)} is (49^{x^3})/(6ln(7))+C