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

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

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

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

2 (3 ln (2) - ln (5))

Decimal

0.94000 \dots

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Take the partial fraction of

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

= 4 \cdot \int_{1}^{2} \frac{1}{x} - \frac{x}{x ^{2} + 1} 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

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

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

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

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

Simplify

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

= 2 (3 ln (2) - ln (5))

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