Class 12 Integrals · Short Answer2 Marksdefinite_integration
Class 12 Maths · Integrals · Short Answer #144
Prove that the convergent improper integral \(\int_0^{\pi/2}(\sqrt{\tan x}+\sqrt{\cot x})dx=\sqrt2\pi\).
✅ Step-by-Step Solution
Final Answer: √2π
- On (0,π/2) both sine and cosine are positive. The sum equals (sin x+cos x)/√(sin x cos x).
- Set u=sin x−cos x, so du=(sin x+cos x)dx and sin x cos x=(1−u²)/2. The limits are −1,1, giving √2∫₋₁¹du/√(1−u²)=√2π.
- Therefore the integral equals √2π.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] On (0,π/2) both sine and cosine are positive. The sum equals (sin x+cos x)/√(sin x cos x).
- [1 mark] Set u=sin x−cos x, so du=(sin x+cos x)dx and sin x cos x=(1−u²)/2. The limits are −1,1, giving √2∫₋₁¹du/√(1−u²)=√2π.
- [0.5 marks] Therefore the integral equals √2π.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int \frac{x \operatorname{asin}{\left(x…
Next Question →Evaluate \(\int_0^{\pi/4}(\sin x+\cos x)/(3+\sin2x)dx\).
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).