NCERT Class 12 · Ch 7 Exercise 7.10 · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.10 Question 20
Evaluate ∫₋π/₂^π/₂(x³+x cos x+tan⁵x+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: The ordinary improper integral does not exist. Its symmetric Cauchy principal value is π (option C under that different interpretation).
- The endpoints are poles of tan⁵x. Each one-sided improper integral must converge separately for the ordinary integral to exist.
- Near π/2, tan⁵x behaves like (π/2−x)^(−5), whose integral diverges to +∞. The opposite endpoint diverges to −∞.
- Symmetric cutoffs cancel the three odd terms exactly and leave an integral of 1 tending to π. This proves the principal value π, not convergence of the original improper integral.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Show ∫₀ᵃf(x)g(x)dx=2∫₀ᵃf(x)dx if f(x)=f(a−x) and g(x)+g(a−x)=4.
Next Question →Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \log{\left(\frac{3…
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).