JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f ( x )\) be a differentiable function defined on \([0,2]\) such that \(f^{\prime}(x)=f^{\prime}(2-x)\) for all \(x \in(0,2),f (0)=1\) and \(f (2)= e ^{2} .\) Then the value of \(\int_{0}^{2} f ( x ) dx\) is ..... .
- A \(1- e ^{2}\)
- B \(1+ e ^{2}\)
- C \(2\left(1- e ^{2}\right)\)
- D \(2\left(1+ e ^{2}\right)\)
Answer & Solution
Correct Answer
(B) \(1+ e ^{2}\)
Step-by-step Solution
Detailed explanation
\(f^{\prime}(x)=f^{\prime}(2-x)\) \(f(x)=-f(2-x)+c\) put \(x=0\) \(f^{\prime}(0)=-f^{\prime}(2)+c\) \(c=f(0)+f(2)=1+e^{2}\) so \(, f(x)+f(2-x)=1+e^{2}\) \(I=\int_{0}^{2} f(x) d x\) \(I=\int_{0}^{2} f(2-x) d x\) \(2 I=\int_{0}^{2}(f(x)+f(2-x)) d x\)…
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