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CBSE Class 12 Math 2023 All Sets Solved Paper

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Question : 1 of 20
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If ddxf(x)=2x+3x\frac{d}{dx} f(x) = 2x + \frac{3}{x} and f(1) = 1, then f(x) is
Solution:  
Given equation is ddxf(x)=2x+3x\frac{d}{dx} f(x) = 2x + \frac{3}{x}
Integrating on both sides
∫ddxf(x) dx=∫(2x+3x)dx\int \frac{d}{dx} f(x) \, dx = \int \left(2x + \frac{3}{x}\right) dx
⇒f(x)=x2+3ln⁡∣x∣+c…(i)\Rightarrow f(x) = x^2 + 3 \ln |x| + c \ldots (i)
Given that f(1) = 1
∴f(1)=1=1+c⇒c=0\therefore f(1) = 1 = 1 + c \Rightarrow c = 0
∴f(x)=x2+3ln⁡∣x∣\therefore f(x) = x^2 + 3 \ln |x|
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