Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Use the table of integrals in this section to find the indefinite integral. $$ \int \frac{2 x}{(1-3 x)^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \frac{2 x}{(1-3 x)^{2}} \) is \( \frac{2}{3(1-3x)} + C \).

Step 1: Identify u and u'

Assign \( u = 1-3x \) and differentiate it to find \( u' \) which gives \( u' = -3 \). Notice that there is a factor of 2 in the numerator of the problem that is not accounted for from \( u' \), we'll address this more in the next step.

Step 2: Rescaling to match integral form

The integral can now be rewritten as \( -\frac{2}{3} \int \frac{-3x}{(1-3x)^2} dx \) from our identification above. Here, we included \( -\frac{2}{3} \) to match \( u' \) with numerator. Now, the integral is in the form \( \int \frac{u' }{u^{2}} dx \).

Step 3: Apply the standard integral formula

The standard formula for this type of integral is \( -\frac{1}{u} + C \). Substituting \( u = 1-3x \) back into the equation, we get \( -\frac{1}{1-3x} \) plus the constant of integration \( C \).

Step 4: Rescale the result

Since we had included the extra factor we now rescale our result to obtain the final answer, i.e. we multiply by \( -\frac{2}{3} \). This results in \( \frac{2}{3(1-3x)}+C \).