KCET · Maths · Indefinite Integration
The value of \( \int \frac{e^{x}(1+x) d x}{\cos ^{2}\left(e^{x} \cdot x\right)} \) is equal to
- A \(-\cot \left(e x^{x}\right)+c \)
- B \( \tan \left(e^{x} \cdot x\right)+c \)
- C \( \tan \left(e^{x}\right)+c \)
- D \( \cot \left(e^{x}\right)+c \)
Answer & Solution
Correct Answer
(B) \( \tan \left(e^{x} \cdot x\right)+c \)
Step-by-step Solution
Detailed explanation
Given that \(\int \frac{e^{x}(1+x) d x}{\cos ^{2}\left(x \cdot e^{x}\right)}\)
Let \(x e^{x}=t\) then, \(e^{x}(1+x) d x=d t\)
Now, \(I=\int \frac{d t}{\cos ^{2} t}=\int \sec ^{2} t d t\)
\(=\tan t=\tan \left(x e^{x}\right)+c\)
Let \(x e^{x}=t\) then, \(e^{x}(1+x) d x=d t\)
Now, \(I=\int \frac{d t}{\cos ^{2} t}=\int \sec ^{2} t d t\)
\(=\tan t=\tan \left(x e^{x}\right)+c\)
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