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INTEGRALS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSINTEGRALS1 Mark
Question
Choose the correct answer in Exercise:
$\int\text{e}^\text{x}\sec\text{x}(1+\tan\text{x})\text{dx}$ equals
  1. $\text{e}^\text{x}\cos\text{x}+\text{C}$
  2. $\text{e}^\text{x}\sec\text{x}+\text{C}$
  3. $\text{e}^\text{x}\sin\text{x}+\text{C}$
  4. $\text{e}^\text{x}\tan\text{x}+\text{C}$
✓

Answer

  1. $\text{e}^\text{x}\sec\text{x}+\text{C}$
$\int\text{e}^\text{x}\sec\text{x}(1+\tan\text{x})\text{dx}$
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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