CBSE Class 12 Mathematics 2026 Main Paper 65/5/3 · Q303 Marksintegration_by_parts
CBSE Class 12 Mathematics 2026 Main Paper 65/5/3 · Question 30 (Integrals)
Evaluate \(\int (x-\sin x)\,dx/(1-\cos x)\). OR Evaluate \(\int_0^2 dx/\sqrt{x^2+2x+3}\).
Alternative 1 3 Marks
Evaluate \(\int (x-\sin x)\,dx/(1-\cos x)\).
Answer: −x cot(x/2)+C.
- Use 1−cos x=2 sin²(x/2) and sin x/(1−cos x)=cot(x/2).
- The derivative of −x cot(x/2) is −cot(x/2)+(x/2)csc²(x/2), the transformed integrand.
- Thus the displayed primitive holds on each interval where the integrand is defined.
Alternative 2 3 Marks
Evaluate \(\int_0^2 dx/\sqrt{x^2+2x+3}\).
Answer: ln((3+√11)/(1+√3)).
- Complete the square to obtain (x+1)²+2 under the root.
- Use the logarithmic primitive ln(x+1+√(x²+2x+3)).
- Evaluate at 2 and 0 and combine the logarithms.