What is the integral from 0 to 4 of arctan(x/4) ?

The integral from 0 to 4 of arctan(x/4) is pi-2ln(2)

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integral from 0 to 4 of arctan(x/4)

\int_{0}^{4} arctan (\frac{x}{4}) d x

π - 2 ln (2)

Decimal

1.75529 \dots

Solution steps

\int_{0}^{4} arctan (\frac{x}{4}) d x

Apply Integration By Parts

= [x arctan (\frac{x}{4}) - \int \frac{4 x}{x ^{2} + 16} d x]_{0}^{4}

\int \frac{4 x}{x ^{2} + 16} d x = 2 ln x^{2} + 16

= [x arctan (\frac{x}{4}) - 2 ln x^{2} + 16]_{0}^{4}

Compute the boundaries:

π - 2 ln (2)

= π - 2 ln (2)

What is the integral from 0 to 4 of arctan(x/4) ?
The integral from 0 to 4 of arctan(x/4) is pi-2ln(2)