Question

Integrate the following with respect to x.

If f'(x) = x + b, f(1) = 5 and f(2) = 13, then find f(x)

Answer 1

f(x) = `int "f'"(x)  "d"x`

f() = `int (x + "b")  "d"x`

= `x^2/2 + "b"x + "c"`

f(1) = 5

⇒ `(1)^2/2 + "b"(1) + "c"` = 5

`1/2 + "b" + "c"` = 5

⇒ b + c = `5 - 1/2`

b + c = `9/2`

⇒ 2b + 2c = 9   .......(1)

f(2) = 13

⇒ `(2)^2/2 + "b"(2) + "c"` = 13

2 + 2b + c = 13

2b + c = 11  ........(2)

Solving equation (1) and (2)

2b + 2c = 9
2b + c = 11
(–)   (–)    (–) 
        c =  – 2

Substitute c = – 2 in equation (2)

2b – 2 = 11

⇒ 2b = 11 + 2

b = `13/2`

f(x) = `x^2/2 + (13x)/2 - 2`