NCERT Class 12 · Ch 7 Exercise 7.3 · Q21
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 21
Integrate sin⁻¹(cos x).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: Let n=⌊(x+π)/(2π)⌋, s=x−2nπ∈[−π,π). An antiderivative is πs/2−s|s|/2+C.
- The principal inverse gives sin⁻¹(cos x)=π/2−|s| after reducing x to s∈[−π,π].
- On −π≤s≤0 integrate π/2+s to get πs/2+s²/2; on 0≤s≤π integrate π/2−s to get πs/2−s²/2.
- These pieces combine into πs/2−s|s|/2. Both values at s=±π are 0, so the primitive joins continuously across periods and has the required derivative.
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int \frac{\cos{\left(2 x \right)}}{\left(\sin{\left(x…
Next Question →Evaluate \(\int \frac{1}{\cos{\left(a - x \right)} \cos{\left(b - x…
All Questions in Integrals — Exercise 7.3
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).