Chapter 5.5, Problem 1PT

To determine

To choose: The appropriate option for the value of ${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx}$ when ${\displaystyle \int f\left(x\right)=F\left(x\right)+C}$.

Answer to Problem 1PT

The appropriate option for the value of ${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx}$ when ${\displaystyle \int f\left(x\right)=F\left(x\right)+C}$ is $\underset{\_}{\text{d)}F\left(g\left(x\right)\right)+C}$.

Explanation of Solution

Formula used:

Substitution rule:

If $u=g\left(x\right)$ is differentiable function whose range is an interval I and f is continuous on I, then ${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx={\displaystyle \int f\left(u\right)du}}$. Here, $u=g\left(x\right)$ and $du=g^{\prime }\left(x\right)dx$.

Calculation:

From the substitution rule,

${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx={\displaystyle \int f\left(u\right)du}}\cdots \left(1\right)$.

In equation (1), if $u=x\text{ and }du=1$, then ${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx={\displaystyle \int f\left(x\right)}}$.

It is known that, ${\displaystyle \int f\left(x\right)=F\left(x\right)+C}$.

Compute the value of ${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx}$ by substituting $u=g\left(x\right)$ and $du=g^{\prime }\left(x\right)$ in equation (1).

$\begin{array}{l}{\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx={\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)}}\\ {\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx=F\left(g\left(x\right)\right)+C}\left[\because {\displaystyle \int f\left(x\right)=F\left(x\right)+C}\right]\end{array}$

Therefore, the appropriate option for the value of ${\displaystyle \int f\left(g\left(x\right)\right)g^{\prime }\left(x\right)dx}$ when ${\displaystyle \int f\left(x\right)=F\left(x\right)+C}$ is $\underset{\_}{\text{d)}F\left(g\left(x\right)\right)+C}$.