NCERT Class 12 · Ch 7 Exercise 7.5 · Q15
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 15
Evaluate \(\int \frac{1}{x^{4} - 1}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\frac{\log{\left(\left|{x - 1}\right| \right)}}{4} - \frac{\log{\left(\left|{x + 1}\right| \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Factor the denominator, perform polynomial division if needed, and solve for the partial-fraction coefficients.
- The integrand rewrites as \(- \frac{1}{2 \left(x^{2} + 1\right)} - \frac{1}{4 \left(x + 1\right)} + \frac{1}{4 \left(x - 1\right)}\). Apply ∫xᵖdx=x^(p+1)/(p+1) for p≠−1, ∫dx/x=log|x|, and the elementary exponential/trigonometric rules.
- An antiderivative is F(x)=\(\frac{\log{\left(\left|{x - 1}\right| \right)}}{4} - \frac{\log{\left(\left|{x + 1}\right| \right)}}{4} - \frac{\operatorname{atan}{\left(x \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{1}{x^{4} - 1}\). Add an arbitrary constant C on each domain component.
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