Ex 7.6

Chapter 7 Class 12 Integrals
Serial order wise

### Transcript

Ex 7.6, 14 ใ๐ฅ(logโก๐ฅ)ใ^2 โซ1โใ๐ฅ(logโก๐ฅ )^2.๐๐ฅ " " ใ โด โซ1โใ๐ฅ(logโก๐ฅ )^2.๐๐ฅใ=โซ1โใ(logโก๐ฅ )^2 ๐ฅ .๐๐ฅใ = (logโก๐ฅ )^2 โซ1โใ๐ฅ .ใ ๐๐ฅโโซ1โ((๐(logโก๐ฅ )^2)/๐๐ฅ โซ1โใ๐ฅ .๐๐ฅใ) ๐๐ฅ = (logโก๐ฅ )^2 . ๐ฅ^2/2โโซ1โ(2(logโก๐ฅ ) 1/๐ฅ โซ1โใ๐ฅ .๐๐ฅใ) ๐๐ฅ Now we know that โซ1โใ๐(๐ฅ) ๐โก(๐ฅ) ใ ๐๐ฅ=๐(๐ฅ) โซ1โ๐(๐ฅ) ๐๐ฅโโซ1โ(๐โฒ(๐ฅ)โซ1โ๐(๐ฅ) ๐๐ฅ) ๐๐ฅ Putting f(x) = x and g(x) = (log x)2 = ๐ฅ^2/2 (logโก๐ฅ )^2โ2โซ1โใlogโก๐ฅ/๐ฅ . ๐ฅ^2/2ใ ๐๐ฅ = ๐ฅ^2/2 (logโก๐ฅ )^2โโซ1โใ๐ฅ logโก๐ฅ ใ ๐๐ฅ Solving I1 I1 = โซ1โใ๐ฅ logโก๐ฅ ใ ๐๐ฅ โซ1โใ๐ฅ logโก๐ฅ ใ ๐๐ฅ=โซ1โ(logโก๐ฅ )๐ฅ ๐๐ฅ =logโก๐ฅ โซ1โ๐ฅ ๐๐ฅโโซ1โ(๐(logโก๐ฅ )/๐๐ฅ โซ1โใ๐ฅ.๐๐ฅใ)๐๐ฅ Now we know that โซ1โใ๐(๐ฅ) ๐โก(๐ฅ) ใ ๐๐ฅ=๐(๐ฅ) โซ1โ๐(๐ฅ) ๐๐ฅโโซ1โ(๐โฒ(๐ฅ)โซ1โ๐(๐ฅ) ๐๐ฅ) ๐๐ฅ Putting f(x) = x and g(x) = log x =logโก๐ฅ (๐ฅ^2/2)โโซ1โใ1/๐ฅ . ๐ฅ^2/2. ๐๐ฅใ =ใ๐ฅ^2/2 logใโกใ ๐ฅใโ1/2 โซ1โใ๐ฅ. ๐๐ฅใ =ใ๐ฅ^2/2 logใโก๐ฅโ1/2 . ๐ฅ^2/2 +๐ถ =ใ๐ฅ^2/2 ๐๐๐ใโกใ ๐ฅใโ ๐ฅ^2/4 +๐ถ Putting value of I1 in (1), โซ1โใ๐ฅ(logโก๐ฅ )^2.๐๐ฅใ=๐ฅ^2/2 (logโก๐ฅ )^2โโซ1โใ ๐ .๐๐๐โก๐ ๐๐ใ =๐ฅ^2/2 (logโก๐ฅ )^2โ((๐ฅ^2 (logโก๐ฅ ))/2 โ ๐ฅ^2/4 +๐ถ1) =๐ฅ^2/2 (logโก๐ฅ )^2โ (๐ฅ^2 (logโก๐ฅ ))/2 + ๐ฅ^2/4 โ๐ถ1 =๐^๐/๐ (๐๐๐โก๐ )^๐โ (๐^๐ (๐๐๐โก๐ ))/๐ + ๐^๐/๐+๐ช " "