What is the integral from 2 to 3 of (x^5+1)/(x^6+6x) ?

The integral from 2 to 3 of (x^5+1)/(x^6+6x) is (5ln(249)+5ln(3)-5ln(2)-5ln(38))/(30)

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integral from 2 to 3 of (x^5+1)/(x^6+6x)

\int_{2}^{3} \frac{x ^{5} + 1}{x ^{6} + 6 x} d x

\frac{5 ln ( 249 ) + 5 ln ( 3 ) - 5 ln ( 2 ) - 5 ln ( 38 )}{30}

Decimal

0.38088 \dots

Solution steps

\int_{2}^{3} \frac{x ^{5} + 1}{x ^{6} + 6 x} d x

Expand

\frac{x ^{5} + 1}{x ^{6} + 6 x} : \frac{x ^{5}}{x ^{6} + 6 x} + \frac{1}{x ^{6} + 6 x}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{2}^{3} \frac{x ^{5}}{x ^{6} + 6 x} d x + \int_{2}^{3} \frac{1}{x ^{6} + 6 x} d x

\int_{2}^{3} \frac{x ^{5}}{x ^{6} + 6 x} d x = \frac{1}{5} (ln (249) - ln (38))

\int_{2}^{3} \frac{1}{x ^{6} + 6 x} d x = \frac{5 ln ( 3 ) - 5 ln ( 2 ) - ln ( 249 ) + ln ( 38 )}{30}

= \frac{1}{5} (ln (249) - ln (38)) + \frac{5 ln ( 3 ) - 5 ln ( 2 ) - ln ( 249 ) + ln ( 38 )}{30}

Simplify

\frac{1}{5} (ln (249) - ln (38)) + \frac{5 ln ( 3 ) - 5 ln ( 2 ) - ln ( 249 ) + ln ( 38 )}{30} : \frac{5 ln ( 249 ) + 5 ln ( 3 ) - 5 ln ( 2 ) - 5 ln ( 38 )}{30}

= \frac{5 ln ( 249 ) + 5 ln ( 3 ) - 5 ln ( 2 ) - 5 ln ( 38 )}{30}

What is the integral from 2 to 3 of (x^5+1)/(x^6+6x) ?
The integral from 2 to 3 of (x^5+1)/(x^6+6x) is (5ln(249)+5ln(3)-5ln(2)-5ln(38))/(30)