What is the integral from 2 to 3 of (x^4+1)/(x^5+5x) ?

The integral from 2 to 3 of (x^4+1)/(x^5+5x) is (-4ln(21)+4ln(3)-4ln(2)+4ln(86))/(20)

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integral from 2 to 3 of (x^4+1)/(x^5+5x)

\int_{2}^{3} \frac{x ^{4} + 1}{x ^{5} + 5 x} d x

\frac{- 4 ln ( 21 ) + 4 ln ( 3 ) - 4 ln ( 2 ) + 4 ln ( 86 )}{20}

Decimal Notation

0.36305 \dots

Solution steps

\int_{2}^{3} \frac{x ^{4} + 1}{x ^{5} + 5 x} d x

Expand

\frac{x ^{4} + 1}{x ^{5} + 5 x} : \frac{x ^{4}}{x ^{5} + 5 x} + \frac{1}{x ^{5} + 5 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 ^{4}}{x ^{5} + 5 x} d x + \int_{2}^{3} \frac{1}{x ^{5} + 5 x} d x

\int_{2}^{3} \frac{x ^{4}}{x ^{5} + 5 x} d x = \frac{1}{4} (ln (86) - ln (21))

\int_{2}^{3} \frac{1}{x ^{5} + 5 x} d x = \frac{4 ln ( 3 ) - 4 ln ( 2 ) - ln ( 86 ) + ln ( 21 )}{20}

= \frac{1}{4} (ln (86) - ln (21)) + \frac{4 ln ( 3 ) - 4 ln ( 2 ) - ln ( 86 ) + ln ( 21 )}{20}

Simplify

\frac{1}{4} (ln (86) - ln (21)) + \frac{4 ln ( 3 ) - 4 ln ( 2 ) - ln ( 86 ) + ln ( 21 )}{20} : \frac{- 4 ln ( 21 ) + 4 ln ( 3 ) - 4 ln ( 2 ) + 4 ln ( 86 )}{20}

= \frac{- 4 ln ( 21 ) + 4 ln ( 3 ) - 4 ln ( 2 ) + 4 ln ( 86 )}{20}

(ln (\frac{1 + x}{1 - x}))^{^{'}}

\int \frac{( x ^{2} - 1 )}{x ^{2} + 1} d x

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

\int sec^{3} (2 x) tan (2 x) d x

x^{2} (1 - x)^{2}

What is the integral from 2 to 3 of (x^4+1)/(x^5+5x) ?
The integral from 2 to 3 of (x^4+1)/(x^5+5x) is (-4ln(21)+4ln(3)-4ln(2)+4ln(86))/(20)