Ex 7.2, 11 - Chapter 7 Class 12 Integrals
Last updated at Dec. 16, 2024 by Teachoo
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Transcript
Ex 7.2, 11 (Method 1)
Integrate the function: ๐ฅ/โ(๐ฅ + 4) , ๐ฅ>0
Step 1:
Simplify the given function
๐ฅ/โ(๐ฅ + 4)
= (๐ฅ + 4 โ 4)/โ(๐ฅ + 4)
= (๐ฅ + 4)/โ(๐ฅ + 4) โ 4/โ(๐ฅ + 4)
= (๐ฅ + 4)^(1/2 โ 1) โ 4(๐ฅ + 4)^(1/2)
= (๐ฅ + 4)^(1/2) โ 4(๐ฅ + 4)^(โ 1/2)
(Adding and Subtracting 4)
Step 2:
Integrating the function
โซ1โใ" " ๐ฅ/โ(๐ฅ + 4)ใ . ๐๐ฅ
= โซ1โ((๐ฅ + 4)^(1/2) " โ " ใ4 (๐ฅ + 4)ใ^(โ 1/2) ) . ๐๐ฅ
= โซ1โ(๐ฅ + 4)^(1/2) . ๐๐ฅ โ 4โซ1โ(๐ฅ + 4)^(โ 1/2) . ๐๐ฅ
= (๐ฅ + 4)^(1/2 + 1)/(1/2 + 1) โ (4 (๐ฅ + 4)^(โ 1/2 + 1))/(โ 1/2 + 1) + C
= (๐ฅ + 4)^(3/2)/(3/2 ) โ (4 (๐ฅ + 4)^(1/2))/( 1/2) + C
= 2/3 (๐ฅ+4)^(3/2) โ 4.2 (๐ฅ+4)^(1/2) + C
= ใ2(๐ฅ+4)ใ^(1/2) ((๐ฅ + 4)/3 โ4) + ๐ถ
= ใ2(๐ฅ+4)ใ^(1/2) ((๐ฅ + 4 โ12)/3) + ๐ถ
= ใ2(๐ฅ+4)ใ^(1/2) ((๐ฅ โ 8))/3 + ๐ถ
= ๐/๐ โ(๐ + ๐) (๐โ๐) + ๐ช
(Taking ใ2(๐ฅ+4)ใ^(1/2) as common)
Ex 7.2, 11 (Method 2)
Integrate the function: ๐ฅ/โ(๐ฅ + 4) , ๐ฅ>0
Step 1:
Let ๐ฅ+4=๐ก
Differentiating both sides ๐ค.๐.๐ก.๐ฅ
1+0= ๐๐ก/๐๐ฅ
1= ๐๐ก/๐๐ฅ
๐๐ฅ=๐๐ก
Step 2:
Integrating the function
โซ1โใ" " ๐ฅ/โ(๐ฅ + 4)ใ . ๐๐ฅ
Putting ๐ก=๐ฅ+4 & ๐๐ฅ=๐๐ก
= โซ1โ๐ฅ/โ๐ก . ๐๐ก
= โซ1โ(๐ก โ 4)/โ๐ก . ๐๐ก
= โซ1โ(๐ก โ 4)/โ๐ก . ๐๐ก
= โซ1โ(๐ก/โ๐ก โ4/โ๐ก) ๐๐ก
= โซ1โ๐ก/โ๐ก . ๐๐ก โ โซ1โ4/โ๐ก . ๐๐ก
= โซ1โ๐ก^(1 โ 1/2) . ๐๐ก โ โซ1โใ4 . ๐ก^(โ 1/2) ใ. ๐๐ก
= โซ1โ๐ก^(1/2) . ๐๐ก โ โซ1โใ4 . ๐ก^(โ 1/2) ใ. ๐๐ก
(As ๐ฅ + 4=๐ก
โ๐ฅ=๐กโ4)
= ๐ก^(1/2 + 1)/(1/2 + 1) โ 4 ๐ก^(โ 1/2 + 1)/(โ 1/2 + 1) +๐ถ
= (๐ก^(3/2) )/(3/2) โ 4 ๐ก^(1/2)/(1/2) +๐ถ
= 2/3 ๐ก^(3/2) โ 4 . 2๐ก^(1/2) +๐ถ
Taking 2/3 . ๐ก^(1/2) as common , we get
= 2/3 . ๐ก^(1/2) (๐กโ4 . 3)+๐ถ
= 2/3 . ๐ก^(1/2) (๐กโ12)+๐ถ
Putting the value of ๐ก=๐ฅ+4
= 2/3 . (๐ฅ+4)^(1/2) (๐ฅ+4โ12)+๐ถ
= ๐/๐ โ(๐+๐) (๐โ๐)+๐ช
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