CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Q325 Markspartial_fractions
CBSE Class 12 Mathematics 2026 Main Paper 65/4/1 · Question 32 (Integrals)
Evaluate \(\int x\,dx/[(x-1)(x^2+4)]\). OR Evaluate \(\int_0^1 x\tan^{-1}x\,dx/(1+x^2)^{3/2}\).
Alternative 1 5 Marks
Evaluate \(\int x\,dx/[(x-1)(x^2+4)]\).
Answer: ln|x−1|/5−ln(x²+4)/10+2 arctan(x/2)/5+C.
- Write the fraction as A/(x−1)+(Bx+C)/(x²+4).
- Matching coefficients gives A=1/5, B=−1/5, C=4/5.
- Integrate the three terms with the logarithm and arctangent primitives.
Alternative 2 5 Marks
Evaluate \(\int_0^1 x\tan^{-1}x\,dx/(1+x^2)^{3/2}\).
Answer: (4−π)/(4√2).
- Substitute x=tan t so the bounds become 0 and π/4.
- The transformed integrand is t sin t. Integrate by parts to obtain −t cos t+sin t.
- Evaluate at the two bounds to get (4−π)/(4√2).