Mathbee CBSE Mathbee · CBSE Class 12 Maths
Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #81

Evaluate \(\int \frac{\left(3 \sin{\left(x \right)} - 2\right) \cos{\left(x \right)}}{- 4 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 5}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(3 \log{\left(2 - \sin{\left(x \right)} \right)} - \frac{4}{\sin{\left(x \right)} - 2}\) + C
  1. Set u=sin x, du=cos x dx. The denominator becomes 5−(1−u²)−4u=(u−2)².
  2. Now (3u−2)/(u−2)²=3/(u−2)+4/(u−2)².
  3. Integrating gives 3ln|u−2|−4/(u−2); substitute u=sin x, noting |sin x−2|=2−sin x.
  4. Therefore the required integral is \(3 \log{\left(2 - \sin{\left(x \right)} \right)} - \frac{4}{\sin{\left(x \right)} - 2}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Set u=sin x, du=cos x dx. The denominator becomes 5−(1−u²)−4u=(u−2)².
  • [1 mark] Now (3u−2)/(u−2)²=3/(u−2)+4/(u−2)².
  • [1 mark] Integrating gives 3ln|u−2|−4/(u−2); substitute u=sin x, noting |sin x−2|=2−sin x.
  • [0.5 marks] Therefore the required integral is \(3 \log{\left(2 - \sin{\left(x \right)} \right)} - \frac{4}{\sin{\left(x \right)} - 2}\) + C.

Master Chapter 7: Integrals

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.
📘 Chapter Notes & All Exercises 🚀 Open Interactive Workspace →
← Previous Question
Evaluate \(\int \frac{x^{2}}{\left(x^{2} + 4\right) \left(x^{2} +…
Next Question →
Evaluate \(\int \frac{\sqrt{x}}{\sqrt{a^{3} - x^{3}}}\,dx\). Work…

More Practice Questions in Integrals

Q1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q27Q28Q29Q30Q31Q32Q33Q34Q35Q36Q37Q38Q39Q40

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 \(\int_{-1}^{1}|x-2|/(x-2)\,dx\).