What is the integral from 0 to pi of arctan(x) ?

The integral from 0 to pi of arctan(x) is piarctan(pi)-1/2 ln(pi^2+1)

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integral from 0 to pi of arctan(x)

\int_{0}^{π} arctan (x) d x

π arctan (π) - \frac{1}{2} ln (π^{2} + 1)

Decimal

2.77367 \dots

Solution steps

\int_{0}^{π} arctan (x) d x

Apply Integration By Parts

: [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{0}^{π}

= [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{0}^{π}

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

= [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{0}^{π}

Simplify

[x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{0}^{π} : [\frac{1}{2} (2 x arctan (x) - ln x^{2} + 1)]_{0}^{π}

= [\frac{1}{2} (2 x arctan (x) - ln x^{2} + 1)]_{0}^{π}

Compute the boundaries:

π arctan (π) - \frac{1}{2} ln (π^{2} + 1)

= π arctan (π) - \frac{1}{2} ln (π^{2} + 1)

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\int_{0}^{1} 6 π x^{2} d x

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

\int_{- 2}^{1} (x + 1)^{3} d x

What is the integral from 0 to pi of arctan(x) ?
The integral from 0 to pi of arctan(x) is piarctan(pi)-1/2 ln(pi^2+1)