Problem 3 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \( y[ \ln|2|- \ln|1/y|\)].

Step 3: 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 \( y*\ln|2y|\).