PDF of $W = X/Y$ where $X,Y$ are continuous random variables having joint PDF $f$? 
Let $(X,Y)$ be an Random Variable of the continuous type with PDF $f$.Let $W =\frac{X}{Y}$.
  Then how do we determine the PDF of $W$ ?

Let $Y = T$ so that $X = WT$ and the determinant of the absolute value of the Jacobian results in $|t|$.
So the PDF of $(W,T)$ is given by $f(w,t) = |t| f(wt,t)$
and as we are interested in calculating the Marginal PDF so integrating with respect to $t$ 
$$f_{W}(w) = \int_{-\infty}^{\infty} |t| f(wt,t) dt = \int_{-\infty}^{\infty} |t| f(wy,y) dy$$
but I wanted them in terms of $x$.Actually the book has this form 
$$f_{W}(w) = \int_{-\infty}^{\infty} |x| f(wx,x) dx$$
Also i thought of getting the required expression perhaps by taking now $x = t$ and then $y = \frac{t}{w}$ resulting in the absolute value of determinant of Jacobian to be $\frac{|t|}{w^2}$ 
so $f(w,t) = \frac{|t|}{w^2} f(t,\frac{t}{w})$
and calculating the marginal PDF
$$f_{W}(w) = \int_{-\infty}^{\infty} \frac{|t|}{w^2} f(t,\frac{t}{w}) dt = \int_{-\infty}^{\infty} \frac{|x|}{w^2} f(x,\frac{x}{w}) dx $$
which is not the expression as given in the book.
Where am I wrong?
Source - An introduction to Probability and statistics by Rohatgi and Saleh (2nd edition,Multiple random variables,functions of random variables ,pg 139,Theorem 3)
 A: After further discussion in the comments and the comments in the OP it seems clear the source of confusion is in the identity
$$
\int_{\mathbb{R}}|x|f_{X,Y}(wx,x) dx = \int_{\mathbb{R}}|y|f_{X,Y}(wy,y)dy
$$
while obviously both are correct, there is the potential for confusion when thinking of the argument in the integrand $x$ as corresponding to the random variable $X$. With that in mind there is clear merit in your choice to leave both expressions in terms of the variable $t$. 
Quick check of the results: let $f_{X,Y}(x,y)$ be the joint density of $(x,y)$ and then taking the transformations
$$
\phi(x,y):\mathbb{R}^2 \rightarrow\mathbb{R}^2;(x,y)\mapsto(x/y,x) \\
\phi^{-1}(w,x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2 ; (w,x)\mapsto(x,x/w) 
$$
I get
$$
\begin{align*}
f_{W,X}(w,x) &= f_{X,Y}(\phi^{-1}(w,x) ) | \mbox{Jac}(\phi^{-1} ) |\\
&= f_{X,Y}(x,x/w) \left| \begin{bmatrix}0 & 1 \\ -x/w^2 & 1/w\end{bmatrix}\right| \\
&= f_{X,Y}(x,x/w) \frac{|x|}{w^2}
\end{align*}
$$
which agrees with you. On the otherhand, and just to demonstrate a slightly different method
$$
\begin{align*}
f_{W}(w) &=\int_\mathbb{R}\int_\mathbb{R} \delta(w - x/y)f_{X,Y}(x,y)dxdy
\end{align*}
$$
and using that for a function with only one root the composition law
$$
\int_{\mathbb{R}}\delta(g(x))f(x) = \frac{f(x_0)}{|g'(x_0)|}, \qquad g(x_0) = 0,
$$
applied to $g(x) = w - x/y$ then
$$
\begin{align*}
f_{W}(w) &=\int_\mathbb{R}\int_\mathbb{R} \delta(w - x/y)f_{X,Y}(x,y)dxdy \\ 
&= \int_{\mathbb{R}} |y| f_{X,Y}(wy, y) dy.
\end{align*}
$$
