Class 12 Integrals · Very Long Answer5 Marksdefinite_integration
Class 12 Maths · Integrals · Very Long Answer #117
Evaluate \(\int_0^{3/2}|x\cos(\pi x)|dx\).
✅ Step-by-Step Solution
Final Answer: 5/(2π)−1/π²
- Since x≥0, split at cosπx=0: x=1/2 and the endpoint 3/2. The product is positive on (0,1/2) and negative on (1/2,3/2).
- Thus I=∫₀^(1/2)x cosπx dx−∫_(1/2)^(3/2)x cosπx dx.
- By parts a primitive is F=x sinπx/π+cosπx/π². Its values are F(0)=1/π², F(1/2)=1/(2π), F(3/2)=−3/(2π).
- The first positive contribution is 1/(2π)−1/π²; the second is 2/π. Add them.
- Therefore the integral equals 5/(2π)−1/π².
📋 CBSE Stepwise Marking Scheme
- [1 mark] Since x≥0, split at cosπx=0: x=1/2 and the endpoint 3/2. The product is positive on (0,1/2) and negative on (1/2,3/2).
- [1 mark] Thus I=∫₀^(1/2)x cosπx dx−∫_(1/2)^(3/2)x cosπx dx.
- [1 mark] By parts a primitive is F=x sinπx/π+cosπx/π². Its values are F(0)=1/π², F(1/2)=1/(2π), F(3/2)=−3/(2π).
- [1 mark] The first positive contribution is 1/(2π)−1/π²; the second is 2/π. Add them.
- [1 mark] Therefore the integral equals 5/(2π)−1/π².
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
For a>0,b>0, evaluate \(\int_0^\pi x/[a^2\cos^2x+b^2\sin^2x]dx\).
Next Question →For \(-\pi/2<\alpha<\pi/2\), evaluate \(\int_0^\pi…
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).