MHT CET · Maths · Indefinite Integration
\(\int \frac{\mathrm{e}^x(1+x)}{\cos ^2\left(\mathrm{e}^x \cdot x\right)} \mathrm{d} x=\)
- A \(-\cot \left(e^x\right)+c\), where \(c\) is a constant of integration.
- B \(\tan \left(x \cdot \mathrm{e}^x\right)+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- C \(\tan \left(\mathrm{e}^x\right)+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- D \(-\cot \left(x \cdot \mathrm{e}^x\right)+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Answer & Solution
Correct Answer
(B) \(\tan \left(x \cdot \mathrm{e}^x\right)+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Step-by-step Solution
Detailed explanation
Let \(\mathrm{I}=\int \frac{\mathrm{e}^x(1+x)}{\cos ^2\left(\mathrm{e}^x \cdot x\right)} \mathrm{d} x\)
Put \(\mathrm{e}^x \cdot x=\mathrm{t} \Rightarrow \mathrm{e}^x(x+1) \mathrm{d} x=\mathrm{dt}\)
\(\mathrm{I} =\int \frac{\mathrm{dt}}{\cos ^2 \mathrm{t}} \)
\( =\int \sec ^2 \mathrm{t} d \)
\( =\tan \mathrm{t}+\mathrm{c} \)
\( \therefore \mathrm{I} =\tan \left(x \cdot \mathrm{e}^x\right)+\mathrm{c}\)
Put \(\mathrm{e}^x \cdot x=\mathrm{t} \Rightarrow \mathrm{e}^x(x+1) \mathrm{d} x=\mathrm{dt}\)
\(\mathrm{I} =\int \frac{\mathrm{dt}}{\cos ^2 \mathrm{t}} \)
\( =\int \sec ^2 \mathrm{t} d \)
\( =\tan \mathrm{t}+\mathrm{c} \)
\( \therefore \mathrm{I} =\tan \left(x \cdot \mathrm{e}^x\right)+\mathrm{c}\)
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