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  • Let <img alt="f(x)=\int \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+3\right)} d x" src="https://entrancecorner.oncodecogs.com/gif.latex?f%28x%29%3D%5Cint%20%5Cfrac%7B2%20x%7D%7B%5Cleft%28x%5E%7B2%7D

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

Option: 1

\log _{e} 19-\log _{e} 20


Option: 2

\log _{e} 17-\log _{e} 18


Option: 3

\frac{1}{2}\left(\log _{e} 19-\log _{e} 17\right)


Option: 4

\frac{1}{2}\left(\log _{e} 17-\log _{e} 19\right)


Answers (1)

best_answer

\text { Let } x^{2}=t
2 x \, d x=d t

f(x)=\int \frac{d t}{(t+1)(t+3)}
=\frac{1}{2} \int\left(\frac{1}{t+1}-\frac{1}{t+3}\right) d t
=\frac{1}{2} \ln \left|\frac{t+1}{t+3}\right|+C

f(x)=\frac{1}{2} \ln \left|\frac{x^{2}+1}{x^{2}+3}\right|+C

x=3
\frac{1}{2} \ln \left(\frac{5}{6}\right)=\frac{1}{2} \ln \left(\frac{5}{6}\right)+C \Rightarrow C=0
\mathrm{f}(\mathrm{x})=\frac{1}{2} \ln \left(\frac{x^{2}+1}{x^{2}+3}\right)
\mathrm{f}(\mathrm{x})=\frac{1}{2} \ln \left(\frac{17}{19}\right)
         =\frac{1}{2}[\ln 17-\ln 19]

Posted by

Divya Prakash Singh

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