NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 10

✅ 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 \(- \frac{24}{5 \left(2 x + 3\right)} + \frac{5}{2 \left(x + 1\right)} - \frac{1}{10 \left(x - 1\right)}\). 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)=\(- \frac{\log{\left(\left|{x - 1}\right| \right)}}{10} + \frac{5 \log{\left(\left|{x + 1}\right| \right)}}{2} - \frac{12 \log{\left(\left|{x + \frac{3}{2}}\right| \right)}}{5}\).
5. Differentiate this expression to check that F′(x)=\(\frac{2 x - 3}{\left(2 x + 3\right) \left(x^{2} - 1\right)}\). Add an arbitrary constant C on each domain component.