CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Q263 Markspartial_fractions
CBSE Class 12 Mathematics 2026 Main Paper 65/3/2 · Question 26 (Integrals)
Evaluate \(\int\frac{\cos x\,dx}{(2+\sin x)(4+\sin x)}\). OR Evaluate \(\int\frac{x+3}{x^2+4x+5}\,dx\).
Alternative 1 3 Marks
Evaluate \(\int\frac{\cos x\,dx}{(2+\sin x)(4+\sin x)}\).
Answer: (1/2)ln((2+sin x)/(4+sin x))+C.
- Substitute t=sin x, so dt=cos x dx.
- Decompose 1/[(2+t)(4+t)]=(1/2)[1/(2+t)−1/(4+t)] and integrate.
- Substitute t=sin x. Both log arguments are positive for real x.
Alternative 2 3 Marks
Evaluate \(\int\frac{x+3}{x^2+4x+5}\,dx\).
Answer: (1/2)ln(x²+4x+5)+arctan(x+2)+C.
- Write x+3=(2x+4)/2+1 and the denominator as (x+2)²+1.
- Integrate the derivative-of-denominator term logarithmically and the remaining term using arctan.
- Combine the two terms and add the integration constant.