Question

Evaluate the partial integral. $$ \int_{1}^{2 y} \frac{y}{x} d x $$

Short answer

The result after evaluating the partial integral is \( y*\ln|2y| \).

Step-by-step solution

Simplify the Integral with Substitution

Let's simplify the problem by substitution a new variable, u, where \(u = x/y\). This gives us the differential of u, \(du = dx/y\), or equivalently \(dx = y du\). Now we substitute those relations into the integral, so \( \int_{1/y}^{2} \frac{1}{u} * y du \). Note that we also change the limits of the integral according to the new variable u.

Evaluate the Simplified Integral

The integral we have now is much simpler to handle. Since the y-term is constant concerning the integration variable we can move it in front of the integral sign. So it transforms into \( y \int_{1/y}^{2} \frac{1}{u} du \), which is a basic form of a logarithm integral. Therefore, we can evaluate it directly to \( y[ \ln|2|- \ln|1/y|\)].

Simplify the Result

Now we can simplify the expressions in the square brackets: \( \ln|2|- \ln|1/y|\) can be combined into \( \ln|2y|\) using the properties of a logarithm. Hence we end up with the result \( y*\ln|2y|\).