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3.7.3 Integration, SPM Practice (Question 5 – 8)


Question 5:
Given  ∫ ( 6 x 2 +1 )dx=m x 3 +x +c,  where m and c are constants, find (a) the value of m. (b) the value of c if  ∫ ( 6 x 2 +1 )dx=13 when x=1.

Solution:
(a)
∫ ( 6 x 2 +1 )dx=m x 3 +x +c 6 x 3 3 +x+c=m x 3 +x+c 2 x 3 +x+c=m x 3 +x+c Compare the both sides, ∴ m=2

(b)

∫ ( 6 x 2 +1 )dx=13 when x=1. 2 ( 1 ) 3 +1+c=13            3+c=13                 c=10

Question 6:
It is given that  ∫ 5 k g(x)dx=6 , and  ∫ 5 k [ g( x )+2 ]dx =14, find the value of k.

Solution:
∫ 5 k [ g( x )+2 ]dx =14 ∫ 5 k g( x )dx + ∫ 5 k 2dx =14                6+ [ 2x ] 5 k =14                 2( k−5 )=8                      k−5=4                           k=9

Question 7:
Given  ∫ k 2 (4x+7)dx=28 , calculate the possible value of k.

Solution:
∫ k 2 (4x+7)dx=28 [ 2 x 2 +7x ] k 2 =28 8+14−( 2 k 2 +7k )=28 22−2 k 2 −7k=28 2 k 2 +7k+6=0 ( 2k+3 )( k+2 )=0 k=− 3 2  or k=−2

Question 8:
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

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