Problem 13 - Essential Calculus

Question

Find the integral. $$ \int \frac{x-3}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

Answer: \(\int \frac{x-3}{x^{2}+1} dx = \ln |x^2 + 1| - 3 \arctan x + C\)

Step 1: Break down the fraction

The first step is to break down this fraction into simpler fractions. We can separate the fraction into two integrals: \[ \int (x-3) dx / (x^2 + 1) = \int x dx / (x^2 + 1) - \int 3 dx / (x^2 + 1) \]

Step 2: Apply the formulas for the integration of quotient

For the first part of the separation, we can notice that the numerator in the integral is the derivative of the denominator. Thus, we apply the integral rule for such case which is, the integral of du/u equals to ln|u|. Therefore, the integral of the first part of the separation will be: \[ \int x dx / (x^2 + 1) = \ln |x^2 + 1| \]. For the second part, we simply apply the formula for integrating the fraction: \[ \int a dx / (x^2 + b) = a \arctan (x / \sqrt{b})\]. Therefore, the integral of the second part will be: \[ \int 3 dx / (x^2 + 1) = 3 \arctan x \].

Step 3: Putting the parts together

Now, we should join together the integrals we have calculated in the previous step to get the answer to the original integral. This results in: \[ \int \frac{x-3}{x^{2}+1} dx = \ln |x^2 + 1| - 3 \arctan x + C \], where C is the constant of integration