What is the indefinite integral \int (3x^2 + 4\cos x) \, dx? — JAMB UTME

What is the indefinite integral $\int (3x^2 + 4\cos x) \, dx$ ?

x^3 + 4\sin x + C

Integrating

3x^2

gives

x^3

via the power rule, and integrating

4\cos x

gives

4\sin x

. Combining these terms along with the required constant of integration

C

results in

x^3 + 4\sin x + 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 = \frac{3x^3}{3} = x^3

The power rule increases the exponent by 1 and divides by the new exponent.

Integrate the trigonometric term

4\cos x

using the standard integral

\int \cos x \, dx = \sin x

\int 4\cos x \, dx = 4\sin x

The antiderivative of cosine is positive sine.

Combine the results and append the constant of integration

C

x^3 + 4\sin x + C

All indefinite integrals must include an arbitrary constant

C

Indefinite Integration of Polynomial and Trigonometric Functions

Question