What is the integral of 65x^{5x}(1+ln(x)) ?

The integral of 65x^{5x}(1+ln(x)) is 13x^{5x}+C

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integral of 65x^{5x}(1+ln(x))

\int 65 x^{5 x} (1 + ln (x)) d x

13 x^{5 x} + C

Solution steps

\int 65 x^{5 x} (1 + ln (x)) d x

Take the constant out:

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

= 65 \cdot \int x^{5 x} (1 + ln (x)) d x

Apply u-substitution

: \int \frac{1}{5} d u

= 65 \cdot \int \frac{1}{5} d u

Integral of a constant:

\int a d x = a x

= 65 \cdot \frac{1}{5} u

Substitute back

u = x^{5 x}

= 65 \cdot \frac{1}{5} x^{5 x}

Simplify

65 \cdot \frac{1}{5} x^{5 x} : 13 x^{5 x}

= 13 x^{5 x}

Add a constant to the solution

= 13 x^{5 x} + C

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What is the integral of 65x^{5x}(1+ln(x)) ?
The integral of 65x^{5x}(1+ln(x)) is 13x^{5x}+C