What is the integral from 0 to 1 of 1/(2+cos(pix)) ?

The integral from 0 to 1 of 1/(2+cos(pix)) is 1/(sqrt(3))

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integral from 0 to 1 of 1/(2+cos(pix))

\int_{0}^{1} \frac{1}{2 + cos ( π x )} d x

\frac{1}{3}

Decimal Notation

0.57735 \dots

Solution steps

\int_{0}^{1} \frac{1}{2 + cos ( π x )} d x

Compute the indefinite integral:

\int \frac{1}{2 + cos ( π x )} d x = \frac{2}{3 π} arctan (\frac{1}{3} tan (\frac{π x}{2})) + C

Compute the boundaries:

\int_{0}^{1} \frac{1}{2 + cos ( π x )} d x = \frac{1}{3} - 0

= \frac{1}{3} - 0

Simplify

= \frac{1}{3}

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What is the integral from 0 to 1 of 1/(2+cos(pix)) ?
The integral from 0 to 1 of 1/(2+cos(pix)) is 1/(sqrt(3))