What is the indefinite integral \int (3x^2 - 5\sin(5x) + 2) \, dx? — JAMB UTME

What is the indefinite integral $\int (3x^2 - 5\sin(5x) + 2) \, dx$ ?

x^3 + \cos(5x) + 2x + C

Integrating term-by-term yields

\int 3x^2 dx = x^3

\int -5\sin(5x) dx = \cos(5x)

, and

\int 2 dx = 2x

. Adding the arbitrary constant

C

produces

x^3 + \cos(5x) + 2x + C

Integrate the polynomial term

3x^2

using the power rule

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

\int 3x^2 dx = 3 \cdot \frac{x^3}{3} = x^3

Applying the power rule for integration.

Integrate the trigonometric term

-5\sin(5x)

using

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

\int -5\sin(5x) dx = -5 \left(-\frac{1}{5}\cos(5x)\right) = \cos(5x)

Integration of the sine function reverses differentiation with a positive sign change for negative sine.

Integrate the constant term

2

and append the constant of integration

C

\int 2 dx = 2x

, giving total antiderivative

x^3 + \cos(5x) + 2x + C

Indefinite integrals always require an arbitrary constant of integration

C

Indefinite Integration of Polynomial and Trigonometric Functions

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