The value of Te+ + 12+ + 3) is equa...
Question

# The value of Te+ + 12+ + 3) is equal to (A) * + In (e* + 1) - in(28* +3) +C (C) x-ın (* + 1) + 1n(28* + 3) +C (B) ** - In (e* + 1) + in (2e + (D) none of these

IIT/JEE
Maths
Solution
115
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( int frac{d x}{left(e^{x}+1right)left(2 e^{x}+3right)} ) 0 Let ( e^{n}=t Rightarrow begin{array}{l}x=log t text { lifinenciating both sides: }-end{array} ) ( frac{d x}{d x}=frac{d log t}{d x} Rightarrow d x=frac{1}{t} d t ) ( f(1): quad int frac{d t}{t(t+1)(2 t+3)} ) Csolving by partial fraction) let ( frac{1}{t(t+1)(2 t+3)}=frac{A}{t}+frac{B}{t+1}+frac{c}{2 t+3} ) [ =Aleft(2 t^{2}+5 t+3right)+Bleft(2 t^{2}+3 tright)+cleft(t^{2}+tright) ] Wh Comparing We tems breh Gides. ( 2 A+2 B+C=0 longrightarrow G ) ( 5 A+3 B+C=0 ) ( 3 A=1 Rightarrow A=1 / 3 ) (3) ( f(4) Rightarrow B=-1, c=frac{4}{3}+A=1 / 3 ) On soluma eqle. ( frac{1}{t(1+t)left(2+t_{3}right)}=frac{1}{3 t}-frac{1}{(1+t)}+frac{4}{3(2 t+3)} ) on sutegenctives boln fides ( rightarrow ) ( int frac{d t}{t(1+t)(2 t+3)}=int frac{d t}{3 t}-int frac{d t}{1+t}+frac{4}{3} int frac{d t}{2 t+3} ) ( =frac{1}{3} log t-log (1+t)+frac{2}{3} log (2 t+3) ) ( =frac{1}{2} x-log left(1+e^{x}right)+frac{2}{3} log left(2 e^{x}+3right)+6 )
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