Question

Integrate the following: ∫84 "latex-formula">

A. 42x
B. 84x
C. 84x + C
D. 42x + C

Answer 1

Answer:

Choice C. $84\, x + C$.

Step-by-step explanation:

Consider the power rule for integration. Let $n$ be a real number that is not equal to $(-1)$. The power rule for integration states that:

$\displaystyle \int x^{n}\, d x = \frac{1}{n + 1}\, x^{n+ 1} + C$,

How could this rule apply to this question, since there's apparently no $x$ (or its powers) in the integrand? Keep in mind that $x^{0} = 1$ for all real (and particularly non-zero) values of $x$. In other words, the integrand $84$ is equal to $84\, x^0$. The integral becomes:

$\displaystyle \int 84\, x^{0}\, dx$.

The constant can be moved outside the integral sign. Therefore:

$\displaystyle \int 84\, x^{0}\, dx= 84 \int x^{0}\, dx$.

Now that resembles the power rule. In particular, $n = 0$, such that $n + 1 = 1$. By the power rule:

$\begin{aligned}84 \int x^{0}\, dx = 84\, \left(\frac{1}{1}\, x^{1} + C\right) = 84\, x + 84\, C\end{aligned}$.

The non-zero constant in front of $C$ can be ignored (where $C$ represents the constant of integration.) Therefore:

$\displaystyle \int 84\, dx = 84\, x + C$.