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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Choose the correct answer in Exercise : $\int\frac{\text{dx}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}$ is equal to
  • ✓
    $\tan^{-1}(\text{e}^{\text{x}})+\text{C}$
  • B
    $\tan^{-1}\text{(e}^{\text{-x}})+\text{C}$
  • C
    $\log(\text{e}^{\text{x}}-\text{e}^{\text{x}})+\text{C}$
  • D
    $\log(\text{e}^{\text{x}}+\text{e}^\text{x})+\text{C}$
✓

Answer

Correct option: A.
$\tan^{-1}(\text{e}^{\text{x}})+\text{C}$
Let $\text{ I}=\int\frac{\text{dx}}{\text{e}^{\text{x}}+\text{e}^{-\text{x}}}\text{dx}=\int\frac{\text{e}^{\text{x}}}{\text{e}^{2\text{x}}+1}\text{dx}$
Also, let $\text{e}^{\text{x}}=\text{t}$
$\Rightarrow\text{e}^{\text{x}}\ \text{dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{1+\text{t}^{2}}$
$=\tan^{-1}\text{t}+\text{C}$
$=\tan^{-1}\text{(e}^{\text{x}})+\text{C}$

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