Chapter 4.8, Problem 13E

To determine

To calculate: The most general antiderivative of the function $g\left(\theta \right)=\cos \theta -5\sin \theta$ and check answer by differentiation.

Answer to Problem 13E

The most general antiderivative of the function $g\left(\theta \right)=\cos \theta -5\sin \theta$ is $F\left(\theta \right)=\sin \theta +5\cos \theta +C$ .

Explanation of Solution

Given information:

The function is given as:

  $g\left(\theta \right)=\cos \theta -5\sin \theta$

Formula used:

The function F is antiderivative of $f\left(x\right)$ on an interval $I$ if $F^{`}\left(x\right)=f\left(x\right)$ for all $x$ in $I$

The function $F$ is known as antiderivative of $f\left(x\right)$ on an interval $I$ the most general antiderivative of the function on interval $I$ is $F\left(x\right)+C$ , where $C$ is an arbitrary constant

Power rule:

  $\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right)=x^n$

Calculation:

Consider the function,

  $g\left(\theta \right)=\cos \theta -5\sin \theta$

The antiderivative of $g\left(\theta \right)$ is,

  $F^{`}\left(\theta \right)=g\left(\theta \right)$

So, by reverse power rule Integrating both sides of equation,

  $\begin{array}{c}{\displaystyle \int F^{`}\left(\theta \right)}={\displaystyle \int f\left(\theta \right)}\\ F\left(\theta \right)={\displaystyle \int f\left(\theta \right)}\end{array}$

Therefore, integrating the function

  $\begin{array}{c}g\left(\theta \right)={\displaystyle \int \cos \theta -5\sin \theta }dx\\ ={\displaystyle \int \cos \theta }d\theta -5{\displaystyle \int \sin \theta d\theta }\\ =\sin \theta +7\cos \theta +C\end{array}$

Thus, the most general antiderivative of the function is $F\left(\theta \right)=\sin \theta +5\cos \theta +C$ .

Now checking the answer by differentiation:

Differentiate the antiderivative function,

  $\begin{array}{c}\frac{d}{dx}F\left(\theta \right)=\frac{d}{dx}\left(\sin \theta +5\cos \theta +C\right)\\ F^{`}\left(\theta \right)=\frac{d}{dx}\left(\sin \theta \right)+\frac{d}{dx}\left(5\cos \theta \right)+\frac{d}{dx}C\end{array}$

further,

  $g\left(\theta \right)=\cos \theta -5\sin \theta$

Thus, answer is correct.