The value of the integral ∫_(1/2)² (tan ⁻¹x)/xdx is equal to :

Solution

Maths sir · 4 steps

Limits 1/2 and 2 are reciprocals, so put x = 1/t and add the two forms of the integral.

1. Substitute x = 1/t. Let I be the integral. With x = 1/t, dx = -1/t² dt, and the limits 1/2 → 2 become 2 → 1/2. The integrand becomes t tan⁻¹ 1/t × (-1/t²) = -tan⁻¹(1/t)/t.
2. Flip the limits. Swapping the limits back to 1/2 → 2 removes the minus sign. So I = ∫_(1/2)² tan⁻¹(1/x)/x dx, writing the dummy variable as x again.

   2I = ∫_(1/2)² (tan⁻¹ x + tan⁻¹ 1/x)/x dx = π/2∫_(1/2)² dx/x

3. Use the identity. For x > 0, tan⁻¹ x + tan⁻¹ 1/x = π/2. Then ∫_(1/2)² dx/x = ln 2 - ln 1/2 = 2 ln 2, so 2I = π/2 × 2 ln 2 = π ln 2.
4. So. I = π/2 logₑ 2, which is option (a).

The idea to keep. When the limits are a and 1/a, try x = 1/t. It keeps the limits and dx/x the same, and tan⁻¹ x + tan⁻¹ 1/x = π/2 does the rest.

Answer from the NTA final answer key. Solution by PrepWiser.