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Recover f and Integrate from f(x²+1)=x⁴+5x²+2 | JEE

JEE Maths question with a full step-by-step solution.

Question
Let ff be a polynomial with f(x2+1)=x4+5x2+2f(x^{2}+1)=x^{4}+5x^{2}+2 for all real xx. Then ∫f(x) dx\displaystyle\int f(x)\,dx is
A$\dfrac{x^{3}}{3}+\dfrac{3x^{2}}{2}-2x+Ccorrect
Bx33+3x22+2x+C\dfrac{x^{3}}{3}+\dfrac{3x^{2}}{2}+2x+C
Cx33−3x22−2x+C\dfrac{x^{3}}{3}-\dfrac{3x^{2}}{2}-2x+C
Dx33−3x22+2x+C\dfrac{x^{3}}{3}-\dfrac{3x^{2}}{2}+2x+C
Solution
Step 1: Substitute t=x2+1t=x^{2}+1, so that x2=t−1x^{2}=t-1 and x4=(x2)2=(t−1)2x^{4}=(x^{2})^{2}=(t-1)^{2}. Then
f(t)=f(x2+1)=x4+5x2+2=(t−1)2+5(t−1)+2f(t)=f(x^{2}+1)=x^{4}+5x^{2}+2=(t-1)^{2}+5(t-1)+2
 Step 2: Expand:
f(t)=(t2−2t+1)+(5t−5)+2=t2+3t−2f(t)=(t^{2}-2t+1)+(5t-5)+2=t^{2}+3t-2
⇒f(x)=x2+3x−2\Rightarrow f(x)=x^{2}+3x-2
 Step 3: Integrate:
∫f(x) dx=∫(x2+3x−2) dx=x33+3x22−2x+C\int f(x)\,dx=\int(x^{2}+3x-2)\,dx=\dfrac{x^{3}}{3}+\dfrac{3x^{2}}{2}-2x+C
 Correct answer: (1)
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