What is the integral from 1 to 5 of (x-arctan(x)) ?

The integral from 1 to 5 of (x-arctan(x)) is 12-5arctan(5)+1/2 ln(26)+pi/4-1/2 ln(2)

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integral from 1 to 5 of (x-arctan(x))

\int_{1}^{5} (x - arctan (x)) d x

12 - 5 arctan (5) + \frac{1}{2} ln (26) + \frac{π}{4} - \frac{1}{2} ln (2)

Decimal Notation

7.20086 \dots

Solution steps

\int_{1}^{5} x - arctan (x) 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}^{5} x d x - \int_{1}^{5} arctan (x) d x

\int_{1}^{5} x d x = 12

\int_{1}^{5} arctan (x) d x = 5 arctan (5) - \frac{1}{2} ln (26) - \frac{π}{4} + \frac{1}{2} ln (2)

= 12 - (5 arctan (5) - \frac{1}{2} ln (26) - \frac{π}{4} + \frac{1}{2} ln (2))

Simplify

= 12 - 5 arctan (5) + \frac{1}{2} ln (26) + \frac{π}{4} - \frac{1}{2} ln (2)

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What is the integral from 1 to 5 of (x-arctan(x)) ?
The integral from 1 to 5 of (x-arctan(x)) is 12-5arctan(5)+1/2 ln(26)+pi/4-1/2 ln(2)