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

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

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

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

\frac{5}{4} ln (2)

Decimal

0.86643 \dots

Solution steps

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

Take the partial fraction of

\frac{x ^{2} + 3}{x ^{3} + 2 x} : \frac{3}{2 x} - \frac{x}{2 ( x ^{2} + 2 )}

= \int_{1}^{2} \frac{3}{2 x} - \frac{x}{2 ( x ^{2} + 2 )} d 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{3}{2 x} d x - \int_{1}^{2} \frac{x}{2 ( x ^{2} + 2 )} d x

\int_{1}^{2} \frac{3}{2 x} d x = \frac{3}{2} ln (2)

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

= \frac{3}{2} ln (2) - \frac{1}{4} ln (2)

Simplify

\frac{3}{2} ln (2) - \frac{1}{4} ln (2) : \frac{5}{4} ln (2)

= \frac{5}{4} ln (2)

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