CBSE 2025 · Region 6 · Set 1 · Q32 · 5 marks
Find : \[\int \frac{\mathrm{x}^{2}+1}{(\mathrm{x}-1)^{2}(\mathrm{x}+3)} d \mathrm{x} \]Evaluate : \[\int_{0}^{\pi / 2} \frac{\mathrm{x}}{\sin \mathrm{x}+\cos \mathrm{x}} d \mathrm{x} \]
Find : \[\int \frac{\mathrm{x}^{2}+1}{(\mathrm{x}-1)^{2}(\mathrm{x}+3)} d \mathrm{x} \]
Evaluate : \[\int_{0}^{\pi / 2} \frac{\mathrm{x}}{\sin \mathrm{x}+\cos \mathrm{x}} d \mathrm{x} \]
Marking-scheme solution
$\displaystyle \frac{\mathrm{x}^{2}+1}{(\mathrm{x}-1)^{2}(\mathrm{x}+3)}=\frac{\mathrm{A}}{\mathrm{x}-1}+\frac{\mathrm{B}}{(\mathrm{x}-1)^{2}}+\frac{\mathrm{C}}{\mathrm{x}+3}=\frac{3 / 8}{\mathrm{x}-1}+\frac{1 / 2}{(\mathrm{x}-1)^{2}}+\frac{5 / 8}{\mathrm{x}+3}$
\[\mathrm{I}=\frac{3}{8} \log |\mathrm{x}-1|-\frac{1}{2(\mathrm{x}-1)}+\frac{5}{8} \log |\mathrm{x}+3|+\mathrm{C}
\]
Let $\displaystyle \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\mathrm{x}}{\sin \mathrm{x}+\cos \mathrm{x}} d \mathrm{x}$
\[\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\dfrac{\pi}{2}-\mathrm{x}}{\cos \mathrm{x}+\sin \mathrm{x}} d \mathrm{x}
\]
using property
$\displaystyle 2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\dfrac{\pi}{2}}{\sin \mathrm{x}+\cos \mathrm{x}} \mathrm{dx}$
\[=\frac{\pi}{2} \frac{1}{\sqrt{2}} \int_{0}^{\frac{\pi}{2}} \frac{1}{\sin \left(\dfrac{\pi}{4}+\mathrm{x}\right)} \mathrm{dx}
\]
\[2 \mathrm{I}=\frac{\pi}{2 \sqrt{2}} \log \left|\operatorname{cosec}\left(\frac{\pi}{4}+\mathrm{x}\right)-\cot \left(\frac{\pi}{4}+\mathrm{x}\right)\right|_{0}^{\frac{\pi}{2}}
\]
IntegralsIntegration by Partial FractionsApplylong_answermedium
More from Integrals
- ∫ (x+5)/((x+6)^2) e^x dx is equal to:2025 · asked 3×
- Evaluate ∫ 1/((e^x+e^-x)(e^x-e^-x)) d x log √22023 · asked 3×
- If ∫ (3 a x)/( b^2+c^2 x^2) d x=A log b^2+c^2 x^2 +K, then the value of A is2026 · asked 3×
- Find: Find: ∫ x^2 log (x^2+1) d x2023 · asked 3×
- If ∫ (2^1/x)/(x^2) d x=k · 2^1/x+C, then k is equal to2025 · asked 3×
- Find: ∫ dx√4 x-x^22022 · asked 3×
- Find: ∫ (2 x)/((x^2+3)(x^2-5)) d x OR Evaluate: ∫ 1^4( x-2 + x-4 ) d x2025 · asked 3×
- Evaluate: ∫ 0^π e^ cos xe^ cos x+e^- cos x d x OR Find: ∫ (2 x+1)/((x+1)^2(x-1)) d x2024 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.