If \int (8x^3 - 9\sin(3x) + 2) \, dx = ax^4 + b\cos(3x) + cx + C, where a, b, and c are co… — JAMB UTME

If $\int (8x^3 - 9\sin(3x) + 2) \, dx = ax^4 + b\cos(3x) + cx + C$ , where $a$ , $b$ , and $c$ are constant coefficients and $C$ is the constant of integration, what is the value of $a + b + c$ ?

Answer: 7

Answer

The value of

a + b + c

is 7.

Integrating term-by-term yields

\int 8x^3 dx = 2x^4

\int -9\sin(3x) dx = 3\cos(3x)

, and

\int 2 dx = 2x

. Equating coefficients with

ax^4 + b\cos(3x) + cx

gives

a = 2

b = 3

, and

c = 2

. Therefore,

a + b + c = 2 + 3 + 2 = 7

Step-by-Step Solution

Integrate the polynomial term

8x^3

\int 8x^3 \, dx = 2x^4

, identifying

a = 2

Apply the power rule of integration:

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

Integrate the trigonometric term

-9\sin(3x)

\int -9\sin(3x) \, dx = 3\cos(3x)

, identifying

b = 3

Apply the standard trigonometric integral formula:

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

Integrate the constant term

2

\int 2 \, dx = 2x

, identifying

c = 2

The integral of a constant

k

with respect to

x

kx

Calculate the requested sum

a + b + c

a + b + c = 2 + 3 + 2 = 7

Summing the coefficients derived from each term's antiderivative.

Key Concept

Indefinite Integration of Polynomial and Trigonometric Functions

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