JEE Mains · Maths · STD 12 - 7.1 indefinite integral
Let \(f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x\) . If \(f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)\), then \(f(4)\) is equal to
- A \(\frac{1}{2}\left(\log _e 17-\log _e 19\right)\)
- B \(\log _e 17-\log _e 18\)
- C \(\frac{1}{2}\left(\log _{ e } 19-\log _{ e } 17\right)\)
- D \(\log _e 19-\log _e 20\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{2}\left(\log _e 17-\log _e 19\right)\)
Step-by-step Solution
Detailed explanation
Put \(x ^2= t\) \(\int \frac{d t}{(t+1)(t+3)}=\frac{1}{2} \int\left(\frac{1}{t+1}-\frac{1}{t+3}\right) d t\) \(f(x)=\frac{1}{2} \ln \left(\frac{x^2+1}{x^2+3}\right)+C\) \(f(3)=\frac{1}{2}(\ln 10-\ln 12)+C\) \(\Rightarrow C=0\) \(f(4)=\frac{1}{2} \ln \left(\frac{17}{19}\right)\)
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