What is the integral from-1 to 0 of (x^{15})/(x^8+1) ?

The integral from-1 to 0 of (x^{15})/(x^8+1) is -1/8+1/8 ln(2)

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integral from-1 to 0 of (x^{15})/(x^8+1)

\int_{- 1}^{0} \frac{x ^{15}}{x ^{8} + 1} d x

- \frac{1}{8} + \frac{1}{8} ln (2)

Decimal Notation

- 0.03835 \dots

Solution steps

\int_{- 1}^{0} \frac{x ^{15}}{x ^{8} + 1} d x

Long division

\frac{x ^{15}}{x ^{8} + 1} : x^{7} - \frac{x ^{7}}{x ^{8} + 1}

= \int_{- 1}^{0} x^{7} - \frac{x ^{7}}{x ^{8} + 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

= \int_{- 1}^{0} x^{7} d x - \int_{- 1}^{0} \frac{x ^{7}}{x ^{8} + 1} d x

\int_{- 1}^{0} x^{7} d x = - \frac{1}{8}

\int_{- 1}^{0} \frac{x ^{7}}{x ^{8} + 1} d x = - \frac{1}{8} ln (2)

= - \frac{1}{8} - (- \frac{1}{8} ln (2))

Simplify

= - \frac{1}{8} + \frac{1}{8} ln (2)

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What is the integral from-1 to 0 of (x^{15})/(x^8+1) ?
The integral from-1 to 0 of (x^{15})/(x^8+1) is -1/8+1/8 ln(2)