What is the integral from 0 to 2 of (5x^2+1)/(x^3+1) ?

The integral from 0 to 2 of (5x^2+1)/(x^3+1) is (21ln(3)+sqrt(3)pi)/6

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integral from 0 to 2 of (5x^2+1)/(x^3+1)

\int_{0}^{2} \frac{5 x ^{2} + 1}{x ^{3} + 1} d x

\frac{21 ln ( 3 ) + 3 π}{6}

Decimal

4.75204 \dots

Solution steps

\int_{0}^{2} \frac{5 x ^{2} + 1}{x ^{3} + 1} d x

Expand

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

Apply the Sum Rule:

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

= \int_{0}^{2} \frac{5 x ^{2}}{x ^{3} + 1} d x + \int_{0}^{2} \frac{1}{x ^{3} + 1} d x

\int_{0}^{2} \frac{5 x ^{2}}{x ^{3} + 1} d x = \frac{10}{3} ln (3)

\int_{0}^{2} \frac{1}{x ^{3} + 1} d x = \frac{ln ( 3 ) + 3 π}{6}

= \frac{10}{3} ln (3) + \frac{ln ( 3 ) + 3 π}{6}

Simplify

\frac{10}{3} ln (3) + \frac{ln ( 3 ) + 3 π}{6} : \frac{21 ln ( 3 ) + 3 π}{6}

= \frac{21 ln ( 3 ) + 3 π}{6}

What is the integral from 0 to 2 of (5x^2+1)/(x^3+1) ?
The integral from 0 to 2 of (5x^2+1)/(x^3+1) is (21ln(3)+sqrt(3)pi)/6