NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 23

✅ Step-by-Step Solution

1. Work for x>0.
2. For x>0, combine the logs as log(1+1/x²). Put u=1+1/x², du=−2dx/x³; the integral is −(1/2)∫u^(1/2)log u du.
3. Integrate by parts with U=log u, dV=u^(1/2)du, giving −u^(3/2)log u/3+2u^(3/2)/9.
4. An antiderivative is F(x)=\(- \frac{\left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}} \log{\left(\left|{1 + \frac{1}{x^{2}}}\right| \right)}}{3} + \frac{2 \left(1 + \frac{1}{x^{2}}\right)^{\frac{3}{2}}}{9}\).
5. Differentiate this expression to check that F′(x)=\(\frac{\sqrt{x^{2} + 1} \left(- 2 \log{\left(x \right)} + \log{\left(x^{2} + 1 \right)}\right)}{x^{4}}\). Add an arbitrary constant C on each domain component.