# Ex 7.7, 12 (Supplementary NCERT) - Chapter 7 Class 12 Integrals

Last updated at April 16, 2024 by Teachoo

Ex 7.7

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Ex 7.7, 2 Important

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Ex 7.7, 12 (Supplementary NCERT) Important Deleted for CBSE Board 2025 Exams You are here

Ex 7.7, 13 (Supplementary NCERT) Deleted for CBSE Board 2025 Exams

Ex 7.7, 14 (Supplementary NCERT) Important Deleted for CBSE Board 2025 Exams

Last updated at April 16, 2024 by Teachoo

Ex 7.7, 12 (Supplementary NCERT) ( + ^2 ) ( + ^2 ) Step 1: It can be written in the form x = A [ / ( + ^2 )]+ B x = A [1+2 ]+ B x = (2A) x + A +B x = (2A) x / = 2A 1 = 2A A = 1/20 = A + B B = A B = ( 1)/2 Thus, x = A [1 + 2x] + B x = 1/2 [1 + 2x] 1/2 Step 2: Integrating the function w.r.t. x 1 ( + ^2 ) = 1 [1/2 [1+2 ] 1/2] ( + ^2 ) = 1 1/2 [1+2 ] ( + ^2 ) 1 1/2 ( + ^2 ) = 1/2 1 (1+2 ) ( + ^2 ) 1/2 1 ( + ^2 ) Solving _ _1= 1/2 1 (1+2 ) ( + ^2 ) . Let + ^2= Diff. both sides w.r.t.x 1 + 2 = / dx = /(1 + 2 ) Thus, our equation becomes As I_1= 1/2 1 (1+2 ) ( + ^2 ) . Putting the value of ( + ^2) and d I_1= 1/2 1 (1+2 ) . /((1 + 2 ) ) I_1= 1/2 1 I_1= 1/2 1 ( )^(1/2) I_1= 1/2 ^( 1/2 ^(+ 1) )/((1/2 + 1) ) + C_1 I_1= 1/2 ^((1 + 2)/2)/(((1 + 2)/2 ) ) + C_1 I_1= 1/2 ^(3/2)/(3/2) + C_1 I_1= ^(3/2)/3 + C_1 I_1= ( ^2 + ) ^(3/2)/3 + C_1 Solving _ I_2= 1/2 1 ( + ^2 ) I_2 = 1/2 1 ( ^2+ ) I_2 = 1/2 1 ( ^2+2( )(1/2) ) I_2 = 1/2 1 ( ^2+2( )(1/2)+(1/2)^2 (1/2)^2 ) I_2 = 1/2 1 (( +1/2)^2 (1/2)^2 ) It is of form 1 ( ^2 ^2 ) = /2 ( ^2 ^2 ) ^2/2 log | + ( ^2 ^2 )|+C_2 Replacing x by (x + 1/2 ) and a by 1/2, we get I_2 = 1/2 [( + 1/2)/2 (( +1/2)^2 (1/2)^2 ) (1/2)^2/2 log | +1/2+ (( +1/2)^2 (1/2)^2 )|+ C_2 ] I_2 = 1/2 [((2 +1)/2)/2 (( +1/2)^2 (1/2)^2 ) (1/2)^2/2 log| +1/2+ (( +1/2)^2 (1/2)^2 )|+ C_2 ] I_2 = 1/2 [(2 +1)/2.2 ( ^2 (1/2)^2+2( )(1/2) (1/2)^2 ) (1/4)/2 log | +1/2+ (( +1/2)^2+2( )(1/2) ) + C_2 ] I_2 = 1/2 [(2 +1)/4 ( ^2+ ) 1/(4 . 2 ) log | +1/2+ (( )^2+2( )(1/2) ) + C_2 ] I_2 = 1/2 [(2 +1)/4 ( ^2+ ) 1/8 log | +1/2+ (( )^2+ )|+ C_2 ] I_2 = ((2 + 1))/8 ( ^2+ ) 1/16 log | +1/2+ (( )^2+ )|+C_3 Putting the value of I_1 and I_2 in (1) , we get 1 ( + ^2 ) = 1/2 1 (1+2 ) ( + ^2 ) 1/2 1 ( + ^2 ) = ( + ^2) ^(3/2)/3+C_1 ((2 + 1) ( ^2 + ))/8+1/16 log | +1/2+ ( ^2+ )| C_3 = ( + ^ ) ^( / )/ (( + ) ( ^ + ))/ + / | + / + ( ^ + )|+