Problem 19 - Essential Calculus

Question

Use integration tables to find the integral. $$ \int \frac{x}{\left(x^{2}-6 x+10\right)^{2}} d x $$

Step-by-Step Solution

Answer

The result of the integral is \( -\frac{1}{2(x^2-6x+10)} + C \).

Step 1: Substitute variables

Firstly, set \( u = x^{2} - 6x + 10 \) . Then, differentiate \( u \) to find \( du \). \( du= (2x-6)dx \) Then, simplify \( du \) to \( du = 2(x-3)dx \). To get \( dx \) alone on one side of the equation, divide both sides by \( 2(x-3) \). So, \( dx = \frac{du}{2(x-3)} \). Substitute these new variables into the initial integral, so that it becomes \( \int \frac{x}{u^{2}} \cdot \frac{du}{2(x-3)} \).

Step 2: Simplify the new integral

To make it simpler, divide \( x \) by \( 2(x-3) \) to get \( \frac{1}{2} \) . Hence the new integral becomes \( \frac{1}{2} \int \frac{du}{u^{2}} \). This is a standard integral and can be solved more easily.

Step 3: Solve the simplified integral

Next step is to integrate \( \frac{1}{2} \int \frac{du}{u^{2}} \). The integral of \( \frac{1}{u^{2}} \) is \( -\frac{1}{u} \). So we get \( -\frac{1}{2u} + C \), where \( C \) is the constant of integration.

Step 4: Substitute \( u \) back to original variable

The final step is to substitute \( u \) back to its original value. So we get \( -\frac{1}{2(x^2-6x+10)} + C \). This is the solution to this integral.