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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
MCQ
$\int\frac{\text{e}^{\text{x}}(1+\text{x})}{\cos^2(\text{x}\text{e}^{\text{x}})}\text{ dx}=$
  • A
    $2\log_\text{e}\cos(\text{xe}^{\text{x}})+\text{C}$
  • B
    $\sec(\text{xe}^{\text{x}})+\text{C}$
  • ✓
    $\tan(\text{xe}^{\text{x}})+\text{C}$
  • D
    $\tan(\text{x}+\text{e}^{\text{x}})+\text{C}$
✓

Answer

Correct option: C.
$\tan(\text{xe}^{\text{x}})+\text{C}$
$\text{I}=\int\frac{\text{e}^{\text{x}}(1+\text{x})}{\cos^{2}(\text{xe}^{\text{x}})}\text{ dx}$
Put $\text{xe}^{\text{x}}=\text{t}$
$\text{e}^{\text{x}}(1+\text{x})\text{dx}=\text{dt}$
$\text{I}=\int\frac{\text{dt}}{\cos^2\text{t}}$
$=\text{I}=\int\sec^2\text{t dt}$
$\text{t}=\tan\text{t}+\text{C}$
$\text{I}=\tan(\text{xe}^{\text{x}})+\text{C}$

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