AP EAMCET · Maths · Indefinite Integration
If \(f(x)\) is anti-derivative of \(g(x)\) and \(\int f(x) g(x)\left(1+f^2(x)\right) d x=F(x)\), then \(F(x)=\)
- A \(\frac{\left.\left(1+f^2 x\right)\right)^2}{4}+C\)
- B \(\frac{\left.\left(1+f^2 x\right)\right)^2}{2}+C\)
- C \(\frac{f^2(x) g(x)}{4}+C\)
- D \(\frac{g^2(x) f(x)}{4}+C\)
Answer & Solution
Correct Answer
(A) \(\frac{\left.\left(1+f^2 x\right)\right)^2}{4}+C\)
Step-by-step Solution
Detailed explanation
Given, \(f(x)=\int g(x) d x\) \(\Rightarrow f^{\prime}(x)=g(x)\) \(F(x)=\int f(x) g(x)\left(1+f^2(x)\right) d x\) On putting \(1+f^2(x)=t\) \(\Rightarrow \quad 2 f(x): f^{\prime}(x) d x=d t\)…
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