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

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

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

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

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

Decimal Notation

0.03835 \dots

Solution steps

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

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

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

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

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