Question

Explain, using the chain rule and / or $u$-substitution, why

$\int \frac{1}{q\left(y\right)}\frac{dy}{dx}dx=\int \frac{1}{q\left(y\right)}dy$.

Short answer

The required result is$\int \frac{1}{q\left(y\right)}\frac{dy}{dx}dx=\int \frac{1}{q\left(y\right)}dy$

Step-by-step solution

Step 1. Given information 

The expression is$\int \frac{1}{q\left(y\right)}\frac{dy}{dx}dx=\int \frac{1}{q\left(y\right)}dy$

Step 2. Calculation

Utilize the substitution $u=y\left(x\right)$and think about the integral on the left side of equation (1). Afterward, using the chain rule of distinction

$\begin{array}{rcl}du& =& d\left[y\right(x\left)\right]\\ & =& y^{\text{'}}\left(x\right)dx\\ & =& \frac{dy}{dx}dx\end{array}$

Replace this in the integral to obtain

$\int \frac{1}{q\left(y\right)}\frac{dy}{dx}dx=\int \frac{1}{q\left(u\right)}du\dots \dots \text{(2)}$

Keep in mind that integration is independent of the integration variable, therefore

$\int f\left(x\right)dx=\int f\left(t\right)dt$

Integrate the preceding result into the right side of equation (2) to obtain

$\int \frac{1}{q\left(u\right)}du=\int \frac{1}{q\left(y\right)}dy\dots ..\left(3\right)$

Therefore, from (2) and (3)

$\int \frac{1}{q\left(y\right)}\frac{dy}{dx}dx=\int \frac{1}{q\left(y\right)}dy$