NCERT Class 12 · Ch 7 Exercise 7.5 · Q12
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.5 Question 12
Evaluate \(\int \frac{x^{3} + x + 1}{x^{2} - 1}\, dx\).
💡 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: \(\frac{x^{2}}{2} + \frac{3 \log{\left(\left|{x - 1}\right| \right)}}{2} + \frac{\log{\left(\left|{x + 1}\right| \right)}}{2}\) + 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 \(x + \frac{1}{2 \left(x + 1\right)} + \frac{3}{2 \left(x - 1\right)}\). Apply ∫xᵖdx=x^(p+1)/(p+1) for p≠−1, ∫dx/x=log|x|, and the elementary exponential/trigonometric rules.
- An antiderivative is F(x)=\(\frac{x^{2}}{2} + \frac{3 \log{\left(\left|{x - 1}\right| \right)}}{2} + \frac{\log{\left(\left|{x + 1}\right| \right)}}{2}\).
- Differentiate this expression to check that F′(x)=\(\frac{x^{3} + x + 1}{x^{2} - 1}\). Add an arbitrary constant C on each domain component.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int \frac{5 x}{\left(x + 1\right) \left(x^{2}…
Next Question →Evaluate \(\int \frac{2}{\left(1 - x\right) \left(x^{2} +…
All Questions in Integrals — Exercise 7.5
Related Questions in Integrals
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int x^2e^{x^3}\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(1+x)\sec^2(xe^x)\,dx\) on an interval where defined.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int e^x(\cos x-\sin x)\,dx\).
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int dx/(\sin^2x\cos^2x)\) on an interval avoiding its poles.
Class 12 Integrals · Mcq Single Correct
If \(\int\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln|4e^x+5e^{-x}|+C\), find a,b.
Class 12 Integrals · Mcq Single Correct
Evaluate \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).