What is the indefinite integral \int \left( 6x^2 - \frac{4}{x^2} + 4\sin(2x)\cos(2x) \righ… — JAMB UTME

What is the indefinite integral $\int \left( 6x^2 - \frac{4}{x^2} + 4\sin(2x)\cos(2x) \right) dx$ ?

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

Simplifying

4\sin(2x)\cos(2x)

2\sin(4x)

using double-angle identities and integrating

6x^2 - 4x^{-2} + 2\sin(4x)

term-by-term yields

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

Use the double-angle trigonometric identity to simplify the product term.

4\sin(2x)\cos(2x) = 2(2\sin(2x)\cos(2x)) = 2\sin(4x)

Applying the double-angle identity

\sin(2\theta) = 2\sin\theta\cos\theta

transforms the product into a standard single sine term.

Express the reciprocal power with a negative exponent.

-\frac{4}{x^2} = -4x^{-2}

Rewriting the fraction in index form allows the power rule of integration to be applied directly.

Integrate each component term using standard integration formulas.

\int 6x^2 dx = 2x^3

\int -4x^{-2} dx = \frac{-4x^{-1}}{-1} = \frac{4}{x}

, and

\int 2\sin(4x) dx = 2\left(-\frac{1}{4}\cos(4x)\right) = -\frac{1}{2}\cos(4x)

Integrating

x^n

gives

\frac{x^{n+1}}{n+1}

and integrating

\sin(ax)

yields

-\frac{1}{a}\cos(ax)

Combine the integrated terms and append the constant of integration.

2x^3 + \frac{4}{x} - \frac{1}{2}\cos(4x) + C

Indefinite integrals require an arbitrary constant

C

to represent the entire family of antiderivatives.

Indefinite Integration of Polynomial and Trigonometric Functions

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