Question

Integral (x pangkat 3 - cos x) dx

Answer 1

You can break the integral of a sum/difference into the sum/difference of the integrals:

$\int (x^3-\cos(x))\;dx = \int x^3\; dx - \int \cos(x)\; dx$

The first integral can be solved using the power rule for integrations:

$\int x^n\; dx = \frac{x^{n+1}}{n+1} \implies \int x^3\; dx = \frac{x^4}{4}$

As for the second, remember the chain of derivation of trigonometric functions:

$\sin(x) \to \cos(x) \to -\sin(x) \to -\cos(x) \to \sin(x) \ldots$

The integration chain is the same, but it follows the opposite direction: the integral of the cosine function is the sine function. So, you have

$- \int \cos(x)\; dx = -\sin(x)$

Since this is an indefinite integral, we must add the generic costant to the answer:

$\frac{x^4}{4} -\sin(x) + C$