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

What is the indefinite integral $\int (5x^4 - 3\sin(x)) \, dx$ ?

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

Integrating

5x^4

gives

x^5

, and integrating

-3\sin(x)

gives

+3\cos(x)

since

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

. Adding the arbitrary constant of integration

C

results in

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

Integrate the polynomial term

5x^4

\int 5x^4 \, dx = \frac{5x^{4+1}}{4+1} = x^5

Apply the power rule for integration:

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

Integrate the trigonometric term

-3\sin(x)

\int -3\sin(x) \, dx = -3(-\cos(x)) = 3\cos(x)

The integral of

\sin(x)

with respect to

x

-\cos(x)

Combine the terms and add the constant of integration

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

An indefinite integral requires an arbitrary constant

C

Indefinite Integration of Polynomial and Trigonometric Functions

Question