Final answer:
To re-write the integrand as the sum of simpler rational functions, use the method of partial fraction decomposition. Factor the denominator, write the integrand as a sum of partial fractions, find the values of constants A and B, and then integrate each term separately using the power rule.
Step-by-step explanation:
To re-write the integrand as the sum of simpler rational functions, we can use the method of partial fraction decomposition. Let's work through the steps:
- Factor the denominator of the integrand, x² - 10x + 21, to obtain (x - 3)(x - 7).
- Write the integrand as a sum of partial fractions:
7x - 1 / (x - 3)(x - 7) = A / (x - 3) + B / (x - 7), where A and B are constants that we need to find. - Now, we need to find the values of A and B. Multiply both sides of the equation by the common denominator (x - 3)(x - 7) to eliminate the denominators:
7x - 1 = A(x - 7) + B(x - 3). - Next, plug in convenient values for x to solve for A and B. One method is to set x = 3, which gives us -20A = -22, so A = 11/10. Setting x = 7 gives us 20B = -8, so B = -2/5.
- Now that we have the values of A and B, we can rewrite the integrand as
7x - 1 / (x - 3)(x - 7) = 11/10 / (x - 3) - 2/5 / (x - 7). - Finally, we can integrate each term separately using the power rule for integration. The integral of 11/10 / (x - 3) dx is 11/10 ln|x - 3| and the integral of -2/5 / (x - 7) dx is -2/5 ln|x - 7| + C, where C is the constant of integration.
Therefore, the indefinite integral ∫ 7x - 1 / (x² - 10x + 21) dx is equal to 11/10 ln|x - 3| - 2/5 ln|x - 7| + C.