If y = \int (4x^3 - 2\sin(x)) \, dx and y = 5 when x = 0, what is the value of the constan… — JAMB UTME

If $y = \int (4x^3 - 2\sin(x)) \, dx$ and $y = 5$ when $x = 0$ , what is the value of the constant of integration $C$ ?

Cevap: 3

The constant of integration

C

is 3.

Integrating

4x^3 - 2\sin(x)

gives

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

. Substituting

x = 0

and

y = 5

yields

5 = 0 + 2(1) + C

, so

C = 3

Integrate the polynomial and trigonometric terms

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

The integral of

4x^3

with respect to

x

x^4

, and the integral of

-2\sin(x)

2\cos(x)

Apply the given initial condition

y(0) = 5

5 = 0^4 + 2\cos(0) + C \implies 5 = 2 + C

Evaluating the expression at

x = 0

uses the trigonometric property

\cos(0) = 1

Solve the linear equation for

C

C = 3

Subtracting 2 from both sides isolates

C

Determining the Constant of Integration from Initial Boundary Conditions

Soru