CBSE Class 12 Sample Paper for 2019 Boards

Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Question 20

Evaluate: Definite Integration -1 → 1  ∫ (x + |x| + 1) / (x + 2|x| + 1)

### Transcript

Question 20 Evaluate: โซ1_(โ1)^1โ(๐ฅ+|๐ฅ|+1)/(๐ฅ^2+2|๐ฅ|+1) We know that |๐ฅ|={โ(โ&๐ฅ, ๐ฅ<0@&๐ฅ, ๐ฅโฅ0)โค So, for โ1 to 0, |x| = โx and for 0 to 1 |x| = x So, our integral becomes โซ1_(โ1)^1โ(๐ฅ+|๐ฅ|+1)/(๐ฅ^2+2|๐ฅ|+1) โซ1_(โ1)^0โ(๐ฅ+|๐ฅ|+1)/(๐ฅ^2+2|๐ฅ|+1) โซ1_0^1โ(๐ฅ+|๐ฅ|+1)/(๐ฅ^2+2|๐ฅ|+1) โซ1_(โ1)^0โ(๐ฅโ๐ฅ+1)/(๐ฅ^2โ2๐ฅ+1) โซ1_0^1โ(๐ฅ+๐ฅ+1)/(๐ฅ^2+2๐ฅ+1) โซ1_(โ1)^0โ1/(๐ฅ^2โ2๐ฅ+1) โซ1_0^1โ(2๐ฅ+1)/(๐ฅ^2+2๐ฅ+1) โซ1_(โ1)^0โ1/(๐ฅโ1)^2 โซ1_0^1โ(2๐ฅ+1)/(๐ฅ+1)^2 โซ1_(โ1)^0โ1/(๐ฅโ1)^2 โซ1_0^1โ(2๐ฅ+1+1โ1)/(๐ฅ+1)^2 โซ1_(โ1)^0โ1/(๐ฅโ1)^2 โซ1_0^1โ(2๐ฅ+2โ1)/(๐ฅ+1)^2 โซ1_(โ1)^0โ1/(๐ฅโ1)^2 โซ1_0^1โ(2(๐ฅ+1)โ1)/(๐ฅ+1)^2 โซ1_(โ1)^0โ1/(๐ฅโ1)^2 โซ1_0^1โ(2(๐ฅ+1) )/(๐ฅ+1)^2 โซ1_0^1โ1/(๐ฅ+1)^2 โซ1_(โ1)^0โ1/(๐ฅโ1)^2 โซ1_0^1โ(2 )/((๐ฅ+1) ) โซ1_0^1โ1/(๐ฅ+1)^2 โซ1_(โ1)^0โ1/(๐ฅโ1)^2 2 โซ1_0^1โ"dx " /((๐ฅ+1) ) โซ1_0^1โ1/(๐ฅ+1)^2 [(๐ฅโ1)^(โ1)/(โ1)]_(โ1)^0 2[lnโกใ|๐ฅ+1|ใ ]_0^1 [(๐ฅ+1)^(โ1)/(โ1)]_0^1 [(โ1)/((๐ฅโ1))]_(โ1)^0 "2" [lnโกใ|๐ฅ+1|ใ ]_0^1 [(โ1)/((๐ฅ+1))]_0^1 [(โ1)/((0โ1))โ(โ1)/((โ1โ1))] 2[lnโกใ|2|ใโlogโกใ|1|ใ ] [(โ1)/((1+1))โ(โ1)/((0+1))] [1โ1/2] 2[lnโกใ|2|ใโ0] [โ1/2+1] 1/2+2 lnโก2โ1/2 2 lnโก2