The rate of change of a function f(x) with respect to x is defined by f'(x) = 3x^2 - 4x + … — JAMB UTME

The rate of change of a function $f(x)$ with respect to $x$ is defined by $f'(x) = 3x^2 - 4x + 6\sin(3x)$ . If $f(0) = 7$ , determine the value of the constant of integration, $C$ .

Answer: 9

Answer

The constant of integration

C

is equal to 9.

Integrating

f'(x) = 3x^2 - 4x + 6\sin(3x)

gives

f(x) = x^3 - 2x^2 - 2\cos(3x) + C

. Substituting

x = 0

and

f(0) = 7

leads to

7 = 0 - 0 - 2(1) + C

, which simplifies to

C = 9

Step-by-Step Solution

Integrate the rate of change function

f'(x) = 3x^2 - 4x + 6\sin(3x)

with respect to

x

f(x) = x^3 - 2x^2 - 2\cos(3x) + C

Using the power rule

\int x^n dx = \frac{x^{n+1}}{n+1}

and trigonometric integration rule

\int \sin(kx) dx = -\frac{1}{k}\cos(kx)

Substitute the initial condition

x = 0

and

f(0) = 7

into the expression for

f(x)

7 = (0)^3 - 2(0)^2 - 2\cos(3 \cdot 0) + C

The curve passes through

x = 0

with value

y = 7

Evaluate the trigonometric term at zero and solve for

C

7 = -2(1) + C \implies C = 9

Since

\cos(0) = 1

, the expression simplifies to

7 = -2 + C

, yielding

C = 9

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions with Boundary Conditions

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