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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Choose the correct option from given four options$:\ \int\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\text{dx}$ is equal to:
  • A
    $\log|1+\cos\text{x}|+\text{C}$
  • B
    $\log|\text{x}+\sin\text{x}|+\text{C}$
  • C
    $\text{x}-\tan\frac{\text{x}}{2}+\text{C}$
  • ✓
    $\text{x}\cdot\tan\frac{\text{x}}{2}+\text{C}$
✓

Answer

Correct option: D.
$\text{x}\cdot\tan\frac{\text{x}}{2}+\text{C}$
$\text{I}=\int\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\text{dx}$
$=\int\frac{​​\text{x}}{1+\cos\text{x}}\text{dx}+\int\frac{\sin\text{x}}{1+\cos\text{x}}\text{dx}$
$=\int\frac{\text{x}}{2\cos^2\frac{\text{x}}{2}}\text{dx}+\int\frac{2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}\text{dx}$
$=\int\text{x}\sec^2\frac{\text{x}}{2}\text{dx}+\int\tan\frac{\text{x}}{2}\text{dx}$
$=\frac{1}{2}\Big[\text{x}\cdot2\tan\frac{\text{x}}{2}-\int2\tan\frac{\text{x}}{2}\text{dx}\Big]+\int\tan\frac{\text{x}}{2}\text{dx}=\text{x}\cdot\tan\frac{\text{x}}{2}+\text{C}$

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