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

The integral from 1 to 2 of (x^4+1)/(x^5+5x) is (4ln(7/2)+4ln(2))/(20)

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

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

\frac{4 ln ( \frac{7}{2} ) + 4 ln ( 2 )}{20}

Decimal

0.38918 \dots

Solution steps

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

\int_{1}^{2} \frac{x ^{4}}{x ^{5} + 5 x} d x = \frac{1}{4} ln (\frac{7}{2})

\int_{1}^{2} \frac{1}{x ^{5} + 5 x} d x = \frac{4 ln ( 2 ) - ln ( \frac{7}{2} )}{20}

= \frac{1}{4} ln (\frac{7}{2}) + \frac{4 ln ( 2 ) - ln ( \frac{7}{2} )}{20}

Simplify

\frac{1}{4} ln (\frac{7}{2}) + \frac{4 ln ( 2 ) - ln ( \frac{7}{2} )}{20} : \frac{4 ln ( \frac{7}{2} ) + 4 ln ( 2 )}{20}

= \frac{4 ln ( \frac{7}{2} ) + 4 ln ( 2 )}{20}

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What is the integral from 1 to 2 of (x^4+1)/(x^5+5x) ?
The integral from 1 to 2 of (x^4+1)/(x^5+5x) is (4ln(7/2)+4ln(2))/(20)