Problem 26 - Essential Calculus

Question

Use integration tables to evaluate the integral. $$ \int_{2}^{4} \frac{x^{2}}{(3 x-5)^{2}} d x $$

Step-by-Step Solution

Answer

After computing the integral, the short answer ends up being \(2/21\).

Step 1: Change of Variables Using Substitution

Firstly, we perform the substitution. Set \(u = 3x - 5\). Differentiate both sides to obtain \(du/dx = 3\) or \(dx = du/3\).

Step 2: Change Integral Limits After Substitution

The initial integral is in the variable \(x\), and its boundaries are \(2, 4\). After substituting \(u\), we need to change the limits as per \(u\). So, when \(x=2\), \(u = 3*2 - 5 =1\) and when \(x=4\), \(u= 3*4 - 5 = 7\). So the new boundaries are \(1\) and \(7\).

Step 3: Substitute into the Integral

Substitute \(u\) and \(du\) into the integral and adjust the boundaries accordingly. Your integral should now look like this: \(1/3 \int_{1}^{7} u^{-2} du\).

Step 4: Compute the Integral

The integral can now be computed. The antiderivative of \(u^{-2}\) is \(-u^{-1}\) or \(-1/u\). So we end up with \(-1/3 [1/u]_{1}^{7}\).

Step 5: Compute the Definite Integral with the New Boundaries

Now replace the boundaries and compute the definite integral. You should end up with \(-1/3 [1/7 - 1/1]\).