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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$:\ \int\limits^{\frac{\pi}{4}}_{\frac{-\pi}{4}}\frac{\text{dx}}{1+\cos2\text{x}}$ is equal to:
  • ✓
    $1$
  • B
    $2$
  • C
    $3$
  • D
    $4$
✓

Answer

Correct option: A.
$1$
We have, $\int\limits^{\frac{\pi}{4}}_{\frac{-\pi}{4}}\frac{\text{dx}}{1+\cos2\text{x}}=\int\limits^{\frac{\pi}{4}}_{\frac{-\pi}{4}}\frac{\text{dx}}{2\cos^2\text{x}}$ $[\because1+\cos2\text{A}=2\cos^2\text{A}]$
$=\frac{1}{2}\int\limits^{\frac{\pi}{4}}_{\frac{-\pi}{4}}\sec^2\text{x dx}$
$=\int\limits^{\frac{\pi}{4}}_0\sec^2\text{x dx}$ $\int\limits^\text{a}_{-\text{a}}\text{f(x)dx}=\begin{cases}\int\limits^\text{a}_0\text{f(x)},&\text{if f(x) is even}\\0,&\text{if f(x) is odd}\end{cases}$
$=\Big[\tan\text{x}\Big]^{\frac{\pi}{4}}_0=1$
$=\tan\frac{\pi}{4}-\tan0$
$=1$

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