What is the integral of 5^{2x^3}2x^2 ?

The integral of 5^{2x^3}2x^2 is (25^{x^3})/(3ln(5))+C

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integral of 5^{2x^3}2x^2

\int 5^{2 x^{3}} 2 x^{2} d x

\frac{2 5 ^{x^{3}}}{3 ln ( 5 )} + C

Solution steps

\int 5^{2 x^{3}} \cdot 2 x^{2} d x

Take the constant out:

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

= 2 \cdot \int 5^{2 x^{3}} x^{2} d x

Apply u-substitution

: \int \frac{2 5 ^{u}}{3} d u

= 2 \cdot \int \frac{2 5 ^{u}}{3} d u

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int 2 5^{u} d u

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

= 2 \cdot \frac{1}{3} \cdot \frac{2 5 ^{u}}{ln ( 25 )}

Substitute back

u = x^{3}

= 2 \cdot \frac{1}{3} \cdot \frac{2 5 ^{x^{3}}}{ln ( 25 )}

Simplify

2 \cdot \frac{1}{3} \cdot \frac{2 5 ^{x^{3}}}{ln ( 25 )} : \frac{2 5 ^{x^{3}}}{3 ln ( 5 )}

= \frac{2 5 ^{x^{3}}}{3 ln ( 5 )}

Add a constant to the solution

= \frac{2 5 ^{x^{3}}}{3 ln ( 5 )} + C

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What is the integral of 5^{2x^3}2x^2 ?
The integral of 5^{2x^3}2x^2 is (25^{x^3})/(3ln(5))+C