Indefinite Integration of Polynomial and Trigonometric Functions Practice Questions — JAMB UTME — Page 2

Question 21Question →

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 $a + b + c$ ?

Show answer & explanation

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

1

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}

.

2

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)

.

3

Integrate the constant term

2

\int 2 \, dx = 2x

, identifying

c = 2

The integral of a constant

k

with respect to

x

is

kx

.

4

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

Question 22Question →

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, $C$ .

Show answer & explanation

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

1

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)

.

2

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

.

3

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

Question 23Question →

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 $k$ ?

Show answer & explanation

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

1

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)

.

2

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.

3

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

Indefinite Integration of Polynomial and Trigonometric Functions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes