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

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

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

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

- \frac{4}{3} + \frac{π}{2} + π

Decimal

3.37905 \dots

Solution steps

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

Expand

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

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

\int_{- 1}^{1} \frac{2}{x ^{2} + 1} d x = π

= - \frac{4}{3} + \frac{π}{2} + π

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