NCERT Class 12 · Ch 7 Exercise 7.9 · Q5
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.9 Question 5
Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} + 1}\, dx\).
💡 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: \(\frac{\pi}{4}\)
- Work on each open interval where the original integrand is real and defined.
- u=cos x, du=−sin x dx.
- An antiderivative is F(x)=\(- \operatorname{atan}{\left(\cos{\left(x \right)} \right)}\).
- At the upper and lower limits, F(b)=\(0\) and F(a)=\(- \frac{\pi}{4}\).
- By the fundamental theorem (using one-sided limits for any integrable endpoint), I=F(b)−F(a)=\(\frac{\pi}{4}\).
Master Chapter 7: Integrals
Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
← Previous Question
Evaluate \(\int\limits_{0}^{2} x \sqrt{x + 2}\, dx\).
Next Question →Evaluate \(\int\limits_{0}^{2} \frac{1}{- x^{2} + x + 4}\, dx\).
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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).