NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 7

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Factor the denominator, perform polynomial division if needed, and solve for the partial-fraction coefficients.
3. The integrand rewrites as \(- \frac{x - 1}{2 \left(x^{2} + 1\right)} + \frac{1}{2 \left(x - 1\right)}\). Apply ∫xᵖdx=x^(p+1)/(p+1) for p≠−1, ∫dx/x=log|x|, and the elementary exponential/trigonometric rules.
4. An antiderivative is F(x)=\(- \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\log{\left(\left|{x - 1}\right| \right)}}{2} + \frac{\operatorname{atan}{\left(x \right)}}{2}\).
5. Differentiate this expression to check that F′(x)=\(\frac{x}{\left(x - 1\right) \left(x^{2} + 1\right)}\). Add an arbitrary constant C on each domain component.