Question

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

Short answer

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

Step-by-step solution

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 \(\int (x + 3x^{2} + 3x^{3} + x^{4}) dx \)

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 \(\int x dx + \int 3x^{2} dx + \int 3x^{3} dx + \int x^{4} dx \)

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

Final answer

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