Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Use partial fractions to find the indefinite integral. $$ \int \frac{10}{x^{2}-10 x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is -ln|x| + ln|x-10| + C

Step 1: Factor the denominator

Factor the denominator to get \(x(x-10)\). So, the integral becomes \( \int \frac{10}{x(x-10)} d x \)

Step 2: Apply Partial Fraction Decomposition

We can decompose the fraction as \( \frac{10}{x(x-10)} = \frac{A}{x} + \frac{B}{x-10} \). To find the values of A and B, multiply both sides of the equation by \( x(x-10) \) to clear out the denominator. So, we get \( 10 = A(x-10) + Bx \). Now equate the coefficients of like terms on both sides. We get A=-1 and B=1.

Step 3: Rewrite the Integrals

Now, rewrite the integrals using the values of A and B we found: \( \int \frac{10}{x(x-10)} d x = \int \frac{-1}{x} d x + \int \frac{1}{x-10} d x \).

Step 4: Evaluate the Integrals

These are both standard forms of integrals. The integral of 1/x is ln|x| and that of 1/(x-a) is ln|x-a|. Therefore, \( \int \frac{-1}{x} d x = -ln|x| \) and \( \int \frac{1}{x-10} d x = ln|x-10| \). Hence, the final integral after adding these together is: -ln|x| + ln|x-10| + C, where C is the constant of integration.