A curve y = F(x) has a gradient function given by \frac{dy}{dx} = 9x^2 - 8x + 6\cos(3x) - … — JAMB UTME

A curve $y = F(x)$ has a gradient function given by $\frac{dy}{dx} = 9x^2 - 8x + 6\cos(3x) - 4\sin(2x)$ . Given that $y(0) = 7$ , determine the value of the constant of integration $C$ when the antiderivative is expressed in the standard form $y = 3x^3 - 4x^2 + 2\sin(3x) + 2\cos(2x) + C$ .

Answer: 5

Answer

The value of the constant of integration

C

is 5.

Integrating

\frac{dy}{dx} = 9x^2 - 8x + 6\cos(3x) - 4\sin(2x)

yields

y = 3x^3 - 4x^2 + 2\sin(3x) + 2\cos(2x) + C

. Substituting

x = 0

into the expression gives

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

. Equating to

y(0) = 7

gives

2 + C = 7

, which solves to

C = 5

Step-by-Step Solution

Integrate the gradient function term-by-term with respect to

x

y = 3x^3 - 4x^2 + 2\sin(3x) + 2\cos(2x) + C

The antiderivative of

9x^2

3x^3

, of

-8x

-4x^2

, of

6\cos(3x)

2\sin(3x)

, and of

-4\sin(2x)

2\cos(2x)

Apply the initial boundary condition

y(0) = 7

3(0)^3 - 4(0)^2 + 2\sin(0) + 2\cos(0) + C = 7 \implies 2 + C = 7

x = 0

\sin(0) = 0

and

\cos(0) = 1

, making the non-zero constant contribution equal to

2(1) = 2

Solve for the constant of integration

C

C = 5

Subtracting 2 from both sides of

2 + C = 7

yields

C = 5

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions with Boundary Conditions

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