Problem 8 - Essential Calculus

Question

Find the indefinite integral. $$ \int \frac{2 x^{2}+7 x-3}{x-2} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int \frac{2x^{2}+7x-3}{x-2} dx\) is \(x^{2} + 11x - 25ln|x-2| + C\), where \(C\) is an arbitrary constant.

Step 1: Polynomial division

First, divide the numerator, \(2x^2 + 7x - 3\), by the denominator, \(x - 2\), using polynomial division. This will result in a new polynomial and a remainder. The division process gives us \(2x + 11\) as quotient and \(-25\) as remainder.

Step 2: Re-write the integral

Rewrite the integral in terms of the quotient and the remainder, which gives us \[\int \frac{2 x^{2}+7 x-3}{x-2} dx = \int (2x + 11 - \frac{25}{x-2}) dx\]

Step 3: Splitting the integral

The next step is to split the integral into three simpler integrals which could be solved individually. So, \[\int (2x + 11 - \frac{25}{x-2}) dx = \int 2x dx + \int 11 dx - \int \frac{25}{x-2} dx\]

Step 4: Solve the integrals

Then, solve each of these integrals individually using the standard integral formulae. \[\int 2x dx = x^{2} + C_1, \int 11 dx = 11x + C_2, -\int \frac{25}{x-2} dx = -25ln|x-2| + C_3\]

Step 5: Combine the results

Finally, add up all the individual results and combine the constants, resulting in the final solution to the integral. \[x^{2} + 11x - 25ln|x-2| + C\] where \(C = C_1 + C_2 + C_3\), is an arbitrary constant due to the nature of indefinite integrals.