JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f(x)=\int_{0}^{x} e^{t} f(t) d t+e^{x}\) be a differentiable function for all \(x \in R\). Then \(f(x)\) equals ..... .
- A \(2 e ^{\left( e ^{ x }-1\right)}-1\)
- B \(e ^{ e ^{ x }}-1\)
- C \(2 e ^{ e ^{ x }}-1\)
- D \(e ^{\left( e ^{ x }-1\right)}\)
Answer & Solution
Correct Answer
(A) \(2 e ^{\left( e ^{ x }-1\right)}-1\)
Step-by-step Solution
Detailed explanation
\(f( x )=\int_{0}^{ x } e ^{ t } f( t ) dt + e ^{ x } \Rightarrow f(0)=1\) differentiating with respect to \(x\) \(f^{\prime}(x)=e^{x} f(x)+e^{x}\) \(f^{\prime}(x)=e^{x}(f(x)+1)\) \(\int_{0}^{x} \frac{f^{\prime}(x)}{f(x)+1} d x=\int_{0}^{x} e^{x} d x\)…
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