Problem 11 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{2 x}{(1-3 x)^{2}} d x $$

Step-by-Step Solution

Answer

The solution to the integral is \( 2/3(1/(1-3x)) + C \)

Step 1: Identify the Integral

The given integral is a basic rational function, which indicates that we might be able to solve it by a simple substitution.

Step 2: Substitution

Let's substitute \( u = 1 - 3x \) for simplicity. The derivative of \( u \) with respect to \( x \) is \( du/dx = -3 \), or \( dx = du/-3 \). Now, substitute \( u \) and \( dx \) into the integral.

Step 3: Use the Integral Table

The integral now becomes \( -2/3 \int \frac{1}{u^2} du \), which is a standard integral that appears in most integral tables. According to the integral table, the integral of \( 1/u^2 \) is \( -1/u \).

Step 4: Back Substitution

Now, go back and substitute \( u = 1 - 3x \) back into the integral to get the final result.

Step 5: Identify Indefinite Integral

After back substitution, we get the final result to be \( -2/3(-1/u) = 2/3(1/(1-3x)) \). Don't forget to add the constant of integration \( C \) to account for the indefinite integral. The final answer is \( 2/3(1/(1-3x)) + C \)