Chapter 5, Problem 1RE

Finding Indefinite Integrals In Exercises 1–14, find the indefinite integral. Check your result by differentiating.

${\displaystyle \int 16dx}$

To determine

To calculate: The indefinite integral ${\displaystyle \int 16dx}$.

Answer to Problem 1RE

Solution:

The indefinite integral ${\displaystyle \int 16dx}$ is $\underset{\_}{16x+C}$.

Explanation of Solution

Given Information:

The provided indefinite integral is ${\displaystyle \int 16dx}$.

Formula used:

The power rule of integrals:

${\displaystyle \int u^{n}du}=\frac{x^{n+1}}{n+1}+C$ $\left(\text{for }n\ne 1\right)$

The power rule of differentiation:

$\frac{d}{du}{u}^n=n{u}^{n-1}+C$

Calculation:

Consider the indefinite integral:

${\displaystyle \int 16dx}$

Rewrite as,

${\displaystyle \int 16x^{0}dx}$

Now apply, the power rule of integrals:

$\begin{array}{c}{\displaystyle \int 16x^{0}dx}=16\left(\frac{x^{0+1}}{0+1}\right)+C\\ =16x+C\end{array}$

Now, check this value by differentiating it using the power rule:

$\begin{array}{c}\frac{d}{dx}\left(16x+C\right)=16+0\\ =16\end{array}$

Since, the value obtained after the derivative matches the integrand. So, it is correct.

Therefore, the indefinite integral ${\displaystyle \int 16dx}$ is $16x+C$.