Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #102
For real n>−1, evaluate \(\int_0^1x^2(1-x)^n dx\).
✅ Step-by-Step Solution
Final Answer: 2/[(n+1)(n+2)(n+3)]
- Set u=1−x to obtain ∫₀¹(1−u)²uⁿdu.
- Expand and integrate the convergent powers: 1/(n+1)−2/(n+2)+1/(n+3).
- Over a common denominator the numerator is (n+2)(n+3)−2(n+1)(n+3)+(n+1)(n+2)=2.
- Therefore the integral equals 2/[(n+1)(n+2)(n+3)].
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Set u=1−x to obtain ∫₀¹(1−u)²uⁿdu.
- [1 mark] Expand and integrate the convergent powers: 1/(n+1)−2/(n+2)+1/(n+3).
- [1 mark] Over a common denominator the numerator is (n+2)(n+3)−2(n+1)(n+3)+(n+1)(n+2)=2.
- [0.5 marks] Therefore the integral equals 2/[(n+1)(n+2)(n+3)].
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int_0^1\operatorname{arccot}(1-x+x^2)dx\), with arccot…
Next Question →Evaluate \(\int_0^\pi e^{2x}\sin(x+\pi/4)dx\).
More Practice Questions in Integrals
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).