Class 12 Integrals · Very Long Answer5 Marksdefinite_integration
Class 12 Maths · Integrals · Very Long Answer #116
For a>0,b>0, evaluate \(\int_0^\pi x/[a^2\cos^2x+b^2\sin^2x]dx\).
✅ Step-by-Step Solution
Final Answer: π²/(2ab)
- Let g=1/(a²cos²x+b²sin²x), positive and symmetric under x↦π−x. Reflection gives I=(π/2)∫₀^πg(x)dx.
- The same symmetry gives ∫₀^πg=2∫₀^(π/2)g. Thus I=π∫₀^(π/2)dx/(a²cos²x+b²sin²x).
- Set u=b tan x. Then du=b sec²x dx, and dividing by cos²x makes the remaining integral (1/b)∫₀^∞du/(a²+u²).
- For a>0 this equals (1/(ab))[arctan(u/a)]₀^∞=π/(2ab); multiply by π.
- Therefore the integral equals π²/(2ab).
📋 CBSE Stepwise Marking Scheme
- [1 mark] Let g=1/(a²cos²x+b²sin²x), positive and symmetric under x↦π−x. Reflection gives I=(π/2)∫₀^πg(x)dx.
- [1 mark] The same symmetry gives ∫₀^πg=2∫₀^(π/2)g. Thus I=π∫₀^(π/2)dx/(a²cos²x+b²sin²x).
- [1 mark] Set u=b tan x. Then du=b sec²x dx, and dividing by cos²x makes the remaining integral (1/b)∫₀^∞du/(a²+u²).
- [1 mark] For a>0 this equals (1/(ab))[arctan(u/a)]₀^∞=π/(2ab); multiply by π.
- [1 mark] Therefore the integral equals π²/(2ab).
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int_0^{\pi/2}x\sin x\cos x/(\sin^4x+\cos^4x)dx\).
Next Question →Evaluate \(\int_0^{3/2}|x\cos(\pi x)|dx\).
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).