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

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

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

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

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

Solution steps

\int x^{3} \cdot 5^{x^{4} + 1} d x

x^{3} 5^{x^{4} + 1} = 5^{1} x^{3} 5^{x^{4}}

= \int 5^{1} x^{3} \cdot 5^{x^{4}} d x

Take the constant out:

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

= 5^{1} \cdot \int x^{3} \cdot 5^{x^{4}} d x

Simplify

= 5 \cdot \int x^{3} \cdot 5^{x^{4}} d x

Apply u-substitution

: \int \frac{1}{4 ln ( 5 )} d u

= 5 \cdot \int \frac{1}{4 ln ( 5 )} d u

Integral of a constant:

\int a d x = a x

= 5 \cdot \frac{1}{4 ln ( 5 )} u

Substitute back

u = 5^{x^{4}}

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

5 \cdot 5^{x^{4}} = 5^{1 + x^{4}}

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

Add a constant to the solution

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

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