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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Choose the correct answer in Exercise$:\ \int\text{e}^\text{x}\sec\text{x}(1+\tan\text{x})\text{dx}$ equals
  • A
    $\text{e}^\text{x}\cos\text{x}+\text{C}$
  • ✓
    $\text{e}^\text{x}\sec\text{x}+\text{C}$
  • C
    $\text{e}^\text{x}\sin\text{x}+\text{C}$
  • D
    $\text{e}^\text{x}\tan\text{x}+\text{C}$
✓

Answer

Correct option: B.
$\text{e}^\text{x}\sec\text{x}+\text{C}$
$\int\text{e}^\text{x}\sec\text{x}(1+\tan\text{x})\text{dx}$
Let $\text{I}=\int\text{e}^\text{x}\sec\text{x}(1+\tan\text{x})\text{dx}=\int\text{e}^\text{x}(\sec\text{x}+\sec\text{x}\tan\text{x})\text{dx}​​$
Also, let $\sec\text{x}=\text{f}(\text{x})$
$\Rightarrow \ \sec\text{x}\tan\text{x}=\text{f}\ '(\text{x})$
It is known that$, \int\text{e}^\text{x}\{\text{f}(\text{x})+\text{f}\ '(\text{x})\}\text{dx}=\text{e}^\text{x}\text{f}(\text{x})+\text{C}$
$\therefore\ \text{I}=\text{e}^\text{x}\sec\text{x}+\text{C}$

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