What is the indefinite integral \int (6x^5 + 8\sin(4x)) \, dx? — JAMB UTME

What is the indefinite integral $\int (6x^5 + 8\sin(4x)) \, dx$ ?

x^6 - 2\cos(4x) + C

Integrating

6x^5

using the power rule gives

\frac{6x^6}{6} = x^6

. Integrating

8\sin(4x)

gives

8 \left(-\frac{\cos(4x)}{4}\right) = -2\cos(4x)

. Summing these and appending the constant of integration

C

yields

x^6 - 2\cos(4x) + C

Split the integral into two individual terms

\int (6x^5 + 8\sin(4x)) \, dx = \int 6x^5 \, dx + \int 8\sin(4x) \, dx

Linearity property of integration allows term-by-term integration.

Integrate the polynomial term

6x^5

\int 6x^5 \, dx = 6 \cdot \frac{x^{5+1}}{5+1} = \frac{6x^6}{6} = x^6

Apply the power rule of integration:

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

Integrate the trigonometric term

8\sin(4x)

\int 8\sin(4x) \, dx = 8 \cdot \left( -\frac{\cos(4x)}{4} \right) = -2\cos(4x)

Apply the standard integral rule:

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

Combine results and add the constant of integration

x^6 - 2\cos(4x) + C

Indefinite integration requires an arbitrary constant

C

Indefinite Integration of Polynomial and Trigonometric Functions

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