What is the indefinite integral \int (8x^3 + 12\cos(4x) - 5) \, dx? — JAMB UTME

What is the indefinite integral $\int (8x^3 + 12\cos(4x) - 5) \, dx$ ?

2x^4 + 3\sin(4x) - 5x + C

Integrating term by term gives

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

\int 12\cos(4x) \, dx = 3\sin(4x)

, and

\int -5 \, dx = -5x

, along with the constant of integration

C

, yielding

2x^4 + 3\sin(4x) - 5x + C

Integrate the polynomial term

8x^3

\frac{8x^{3+1}}{3+1} = \frac{8x^4}{4} = 2x^4

Apply the power rule for integration:

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

Integrate the trigonometric term

12\cos(4x)

12 \cdot \frac{\sin(4x)}{4} = 3\sin(4x)

Apply the standard trigonometric integral rule:

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

Integrate the constant term

-5

-5x

The integral of a constant

k

with respect to

x

kx

Combine all integrated terms and add the arbitrary constant of integration

2x^4 + 3\sin(4x) - 5x + C

An indefinite integral represents a family of functions and requires the addition of

+ C

Indefinite integration of polynomial and trigonometric functions

Estimated Time:1m 30s

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