Given that \int \left( k x^3 + 12\cos(3x) \right) dx = 4x^4 + 4\sin(3x) + C, where C is th… — JAMB UTME

Given that $\int \left( k x^3 + 12\cos(3x) \right) dx = 4x^4 + 4\sin(3x) + C$ , where $C$ is the arbitrary constant of integration, what is the numerical value of the constant $k$ ?

Answer: 16

Answer

The numerical value of the constant

k

is 16.

Integrating

kx^3 + 12\cos(3x)

with respect to

x

yields

\frac{k}{4}x^4 + 4\sin(3x) + C

. Comparing the coefficient of

x^4

with the given result

4x^4 + 4\sin(3x) + C

gives

\frac{k}{4} = 4

, which leads to

k = 16

Step-by-Step Solution

Integrate the polynomial and trigonometric terms separately using standard integration rules.

\int (kx^3 + 12\cos(3x)) dx = \frac{k}{4}x^4 + 4\sin(3x) + C

Applying the power rule

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

gives

\int kx^3 dx = \frac{k}{4}x^4

, and applying

\int \cos(ax) dx = \frac{\sin(ax)}{a}

gives

\int 12\cos(3x) dx = \frac{12}{3}\sin(3x) = 4\sin(3x)

Equate the integrated expression to the right-hand side of the given equation.

\frac{k}{4}x^4 + 4\sin(3x) + C = 4x^4 + 4\sin(3x) + C

Both sides represent the same antiderivative of the function.

Equate corresponding coefficients of

x^4

to solve for

k

k = 16

\frac{k}{4} = 4 \implies k = 4 \times 4 = 16.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions

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