Integration, Short Questions (Question 2 – 4)


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

Solution:
4 ( 1 + x ) 4 d x = m ( 1 + x ) n + c 4 ( 1 + x ) 4 d x = 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 [ m x + 3 g ( x ) ] d x = 15 1 2 m x d x + 1 2 3 g ( x ) d x = 15 [ m x 2 2 ] 1 2 + 3 1 2 g ( x ) d x = 15 [ m ( 2 ) 2 2 m ( 1 ) 2 2 ] + 3 2 1 2 2 g ( x ) d x = 15 2 m 1 2 m + 3 2 ( 4 ) = 15 given 1 2 2 g ( x ) d x = 4 3 2 m + 6 = 15 3 2 m = 9 m = 9 × 2 3 m = 6

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

Solution:
Given d d x ( 2 x 3 x ) = g ( x ) g ( x ) d x = 2 x 3 x Thus, 1 2 g ( x ) d x = [ 2 x 3 x ] 1 2 = 2 ( 2 ) 3 2 2 ( 1 ) 3 1 = 4 1 = 3

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