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Short Question 11 – 13

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Question 11:
Given  ∫ −2 3 g(x)dx=4 , and  ∫ −2 3 h(x)dx=9 , find the value of (a)  ∫ −2 3 5g(x)dx, (b) m if  ∫ −2 3 [ g(x)+3h( x )+4m ]dx=12

Solution:
(a)
∫ −2 3 5g(x)dx=5 ∫ −2 3 g(x)dx                  =5×4                  =20

(b)
∫ −2 3 [ g(x)+3h( x )+4m ]dx=12 ∫ −2 3 g(x)dx+3 ∫ −2 3 h( x )dx+ ∫ −2 3 4mdx=12 4+3( 9 )+4m [ x ] −2 3 =12        4m[ 3−( −2 ) ]=−19                       20m=−19                           m=− 19 20



Question 12:
(a) Find the value of  ∫ −1 1 ( 3x+1 ) 3 dx. (b) Evaluate  ∫ 3 4 1 2x−4  dx.

Solution:
a)  ∫ −1 1 ( 3x+1 ) 3 dx=[ ( 3x+1 ) 4 4( 3 ) ] −1 1                            = [ ( 3x+1 ) 4 12 ] −1 1                            = 1 12 [ 4 4 − ( −2 ) 4 ]                            = 1 12 ( 256−16 )                            =20

(b)  ∫ 3 4 1 2x−4  dx= ∫ 3 4 1 ( 2x−4 ) 1 2  dx                             = ∫ 3 4 ( 2x−4 ) − 1 2  dx                             = [ ( 2x−4 ) − 1 2 +1 1 2 ( 2 ) ] 3 4                             = [ 2x−4 ] 3 4                             =[ 2( 4 )−4 − 2( 3 )−4 ]                             =2− 2



Question 13:
Given that y= x 2 2x−1 , show that dy dx = 2x( x−1 ) ( 2x−1 ) 2 . Hence, evaluate  ∫ −2 2 x( x−1 ) 4 ( 2x−1 ) 2  dx .

Solution:
y= x 2 2x−1 dy dx = ( 2x−1 )( 2x )−x( 2 ) ( 2x−1 ) 2     = 4 x 2 −2x−2 x 2 ( 2x−1 ) 2     = 2 x 2 −2x ( 2x−1 ) 2     = 2x( x−1 ) ( 2x−1 ) 2  ( shown ) ∫ −2 2 2x( x−1 ) ( 2x−1 ) 2  dx = [ x 2 2x−1 ] −2 2 1 8 ∫ −2 2 2x( x−1 ) ( 2x−1 ) 2  dx = 1 8 [ x 2 2x−1 ] −2 2 1 4 ∫ −2 2 x( x−1 ) ( 2x−1 ) 2  dx = 1 8 [ ( 2 2 2( 2 )−1 )−( ( −2 ) 2 2( −2 )−1 ) ]                            = 1 8 [ ( 4 3 )−( 4 −5 ) ]                            = 1 8 ( 32 15 )                            = 4 15