Find the exact value of ∫(from 0 to 1) x tan⁻¹x dx.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [1] Begin integration by parts and obtain px² tan⁻¹ x + q ∫ x²/(1+x²) dx. Condone sign error in formula for integration by parts.
2. [1] Obtain (1/2)x² tan⁻¹x - (1/2) ∫ x²/(1+x²) dx. OE. Allow with arctanx or tan⁻¹x.
3. [1] Split the fraction x²/(1+x²) = 1 - 1/(1+x²) and integrate to obtain βx + γ tan⁻¹x.
4. [1] Obtain (1/2)x² tan⁻¹x - (1/2)(x - tan⁻¹x). OE.
5. [1] Substitute correct limits correctly in an expression of the form αx² tan⁻¹x + βx + γ tan⁻¹x. Needs to see evidence of the use of the lower limit at least once. Must evaluate the trigonometry using radians.
6. [1] Obtain π/4 - 1/2. ISW. Or exact two term equivalent.