Question

Evaluate the partial integral. $$ \int_{x}^{x^{2}} \frac{y}{x} d y $$

Short answer

The value of the given definite integral is \(\frac{x^{3}}{2} - \frac{x}{2}\)

Step-by-step solution

Define the Integral Function

First, realize the problem is to evaluate the integral \(\int_{x}^{x^{2}} \frac{y}{x} dy\). The function to be integrated is \(f(y) = \frac{y}{x}\), and the limits of the integral are \(x\) (lower limit) and \(x^{2}\) (upper limit). Note that 'x' is considered as a constant with respect to 'y' as our integral is with respect to 'y'.

Integration Process

Now, integrate function \(f(y) = \frac{y}{x}\). This becomes \(\int \frac{y}{x} dy\). Since 'x' can be considered constant in terms of 'y', the integral becomes \(\frac{1}{x} \int y dy\), which after integrating is \(\frac{y^{2}}{2x}\).

Application of Limits

Next, apply the limits to the integral, from 'x' to \(x^{2}\). This becomes \(\frac{1}{2x}[ (x^{2})^{2} - x^{2} ] =\frac{x^{4} - x^{2}}{2x}=\frac{x^{3}}{2} - \frac{x}{2}\).

Simplify

Finally, simply the result to get the answer. In this case, we don't need to simplify it anymore.