Question

V.x² +1{log(x² +1) – 2 log x} oxy dx : 111. Evaluate pe

SSC CGL

Maths

Solution

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( I=int frac{sqrt{x^{2}+1}left{log left(x^{2}+1right)-2 log xright}}{x^{4}} d x ) ( Rightarrow I=frac{-1}{2} int sqrt{t} log (t) d t ) ( Rightarrow-2 I=int frac{sqrt{t}}{pi} cdot frac{log t}{1} d l ) ( Rightarrow-2 I=log t . int sqrt{t} d t-int frac{1}{t}left(int sqrt{t} d tright) d t ) ( begin{aligned} Rightarrow-2 I &=frac{2 t^{3 / 2}}{3} log t-int frac{2 t^{3 / 2}}{3 t} d t Rightarrow-I &=frac{t^{3 / 2} log t-frac{1}{3} int t^{1 / 2} d t}{frac{t^{3}}{3} log t-frac{2 t^{3 / 2}}{9}+c} end{aligned} ) ( Rightarrow quad I=frac{t^{3 / 2}}{9}(2-3 log t)+c ) bleme ( t=1+frac{1}{x^{2}} )