Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Use partial fractions to find the indefinite integral. $$ \int \frac{x^{4}}{(x-1)^{3}} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{2} x^{2} + x^{3} + \frac{3}{4} x^{4} + \frac{1}{5} x^{5} + C\)

Step 1: Polynomial division

Firstly, check if the degree of the numerator of the function inside the integral is equal or greater than the degree of the denominator. If it is, perform a polynomial division to simplify the function. In this case, \(x^{4}\) divided by \((x - 1)^{3}\) gives \(x^{4} + 3x^{3} + 3x^{2} + x\). So, the function under the integral can be rewritten as \(\int (x + 3x^{2} + 3x^{3} + x^{4}) dx \)

Step 2: Integral of polynomial function

Now proceed to integrate each term separately. The integral of a sum or difference of functions is equivalent to the sum or difference of their respective integrals. So the integral can be expressed as \(\int x dx + \int 3x^{2} dx + \int 3x^{3} dx + \int x^{4} dx \)

Step 3: Applying the Power Rule

Using the power rule for integrals, which is \(\int x^{n} dx = \frac{1}{n + 1} x^{n + 1}\) for any real number \(n\) that is not equal to -1, find the antiderivative for each of the four integrals. This yields \(\frac{1}{2} x^{2} + x^{3} + \frac{3}{4} x^{4} + \frac{1}{5} x^{5}\)

Step 4: Final answer

Include the constant of integration \(C\) at the end to account for the indefinite integral which gives the final solution as: \(\frac{1}{2} x^{2} + x^{3} + \frac{3}{4} x^{4} + \frac{1}{5} x^{5} + C\)