Integration, Short Questions (Question 2 - 4)


Question 2:
Given that  4 ( 1+x ) 4 dx=m ( 1+x ) n +c,
find the values of m and n.

Solution:
4 ( 1+x ) 4 dx=m ( 1+x ) n +c 4 ( 1+x ) 4 dx=m ( 1+x ) n +c 4 ( 1+x ) 3 3( 1 ) +c=m ( 1+x ) n +c 4 3 ( 1+x ) 3 +c=m ( 1+x ) n +c m= 4 3 , n=3


Question 3:
Given  1 2 2g(x)dx=4 , and  1 2 [ mx+3g( x ) ]dx =15. Find the value of constant m.

Solution:
1 2 [ mx+3g( x ) ]dx =15 1 2 mxdx + 1 2 3g( x )dx =15 [ m x 2 2 ] 1 2 +3 1 2 g( x )dx =15 [ m ( 2 ) 2 2 m ( 1 ) 2 2 ]+ 3 2 1 2 2g( x )dx =15 2m 1 2 m+ 3 2 ( 4 )=15 given  1 2 2g(x)dx=4 3 2 m+6=15 3 2 m=9 m=9× 2 3 m=6

Question 4:
Given  d dx ( 2x 3x )=g( x ), find  1 2 g(x)dx.

Solution:
Given  d dx ( 2x 3x )=g( x ) g( x )dx= 2x 3x Thus, 1 2 g(x)dx = [ 2x 3x ] 1 2                = 2( 2 ) 32 2( 1 ) 31                =41                =3