Given that F(x) = \int (3x^2 - 2\sin x) \, dx and F(0) = 6, what is the value of the const… — JAMB UTME

Given that $F(x) = \int (3x^2 - 2\sin x) \, dx$ and $F(0) = 6$ , what is the value of the constant of integration $C$ ?

Answer: 4

The value of the constant of integration

C

4

Integrating

3x^2 - 2\sin x

yields

F(x) = x^3 + 2\cos x + C

. Substituting

x = 0

gives

F(0) = 2(1) + C = 2 + C

. Since

F(0) = 6

, setting

2 + C = 6

yields

C = 4

Integrate the function

f(x) = 3x^2 - 2\sin x

with respect to

x

F(x) = x^3 + 2\cos x + C

The antiderivative of

3x^2

x^3

and the antiderivative of

-2\sin x

2\cos x

Evaluate

F(0)

using the antiderivative expression

F(0) = 0^3 + 2\cos(0) + C = 2 + C

Since

\cos(0) = 1

, the term

2\cos(0)

simplifies to

2

Solve for the integration constant

C

using

F(0) = 6

C = 4

Subtracting

2

from both sides of

2 + C = 6

yields

C = 4

Indefinite integration of polynomial and trigonometric functions with initial conditions

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