What is the integral of 5^{x^3}8x^2 ?

The integral of 5^{x^3}8x^2 is 8/(3ln(5))*5^{x^3}+C

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integral of 5^{x^3}8x^2

\int 5^{x^{3}} 8 x^{2} d x

\frac{8}{3 ln ( 5 )} \cdot 5^{x^{3}} + C

Solution steps

\int 5^{x^{3}} \cdot 8 x^{2} d x

Take the constant out:

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

= 8 \cdot \int 5^{x^{3}} x^{2} d x

Apply u-substitution

: \int \frac{1}{3 ln ( 5 )} d u

= 8 \cdot \int \frac{1}{3 ln ( 5 )} d u

Integral of a constant:

\int a d x = a x

= 8 \cdot \frac{1}{3 ln ( 5 )} u

Substitute back

u = 5^{x^{3}}

= 8 \cdot \frac{1}{3 ln ( 5 )} \cdot 5^{x^{3}}

Simplify

8 \cdot \frac{1}{3 ln ( 5 )} \cdot 5^{x^{3}} : \frac{8}{3 ln ( 5 )} \cdot 5^{x^{3}}

= \frac{8}{3 ln ( 5 )} \cdot 5^{x^{3}}

Add a constant to the solution

= \frac{8}{3 ln ( 5 )} \cdot 5^{x^{3}} + C

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What is the integral of 5^{x^3}8x^2 ?
The integral of 5^{x^3}8x^2 is 8/(3ln(5))*5^{x^3}+C