Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #105
Evaluate \(\int_0^\pi x\tan x/(\sec x+\tan x)dx\), using the continuous extension at x=π/2.
✅ Step-by-Step Solution
Final Answer: π(π−2)/2
- Where defined, tan x/(sec x+tan x)=sin x/(1+sin x), which extends continuously to all of [0,π]. Let g(x)=sin x/(1+sin x).
- Since g(π−x)=g(x), reflection gives I=(π/2)∫₀^πg(x)dx.
- A nonsingular primitive of g is x+cos x/(1+sin x). Its endpoint difference is (π−1)−1=π−2.
- Therefore the integral equals π(π−2)/2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Where defined, tan x/(sec x+tan x)=sin x/(1+sin x), which extends continuously to all of [0,π]. Let g(x)=sin x/(1+sin x).
- [1 mark] Since g(π−x)=g(x), reflection gives I=(π/2)∫₀^πg(x)dx.
- [1 mark] A nonsingular primitive of g is x+cos x/(1+sin x). Its endpoint difference is (π−1)−1=π−2.
- [0.5 marks] Therefore the integral equals π(π−2)/2.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int_{-2}^2x^2/(1+5^x)dx\).
Next Question →Evaluate \(\int_0^{\pi/4}(\sin x+\cos x)/(16+9\sin2x)dx\).
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).