What is the integral of 5^{x^4+2x}(2x^3+1) ?

The integral of 5^{x^4+2x}(2x^3+1) is (5^{x^4+2x})/(2ln(5))+C

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integral of 5^{x^4+2x}(2x^3+1)

\int 5^{x^{4} + 2 x} (2 x^{3} + 1) d x

\frac{5 ^{x^{4} + 2 x}}{2 ln ( 5 )} + C

Solution steps

\int 5^{x^{4} + 2 x} (2 x^{3} + 1) d x

Apply u-substitution

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

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

Take the constant out:

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

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

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

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

Substitute back

u = x^{4} + 2 x

= \frac{1}{2} \cdot \frac{5 ^{x^{4} + 2 x}}{ln ( 5 )}

Simplify

\frac{1}{2} \cdot \frac{5 ^{x^{4} + 2 x}}{ln ( 5 )} : \frac{5 ^{x^{4} + 2 x}}{2 ln ( 5 )}

= \frac{5 ^{x^{4} + 2 x}}{2 ln ( 5 )}

Add a constant to the solution

= \frac{5 ^{x^{4} + 2 x}}{2 ln ( 5 )} + C

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What is the integral of 5^{x^4+2x}(2x^3+1) ?
The integral of 5^{x^4+2x}(2x^3+1) is (5^{x^4+2x})/(2ln(5))+C