Determine the function f ( x ) .

            f ′ ( x )= 4x³ - 9 + 2 sin x + 7e^x , f (0) = 15

Solution

The first step is to integrate to fine out the most general possible

f ( x ) = ∫ 4x³ - 9 + 2 sin x + 7e^x

= x⁴ - 9 x - 2 cos x + 7e^x + c

Now we contain a value of the function therefore let's plug in x = 0 and determine the value of the constant of integration c.

15 = f (0) = 0⁴ - 9 (0) - 2 cos (0) + 7e⁰ + c

= -2 + 7 + c

= 5 +c

Thus, from this it looks like c = 10 .  it means that the function is,

f ( x ) = x⁴ - 9x - 2 cos x + 7e^x + 10