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Chapter 12: Problem 30

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

Short Answer

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

Step by step solution

01

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 \(\int (x + 3x^{2} + 3x^{3} + x^{4}) dx \)
02

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 \(\int x dx + \int 3x^{2} dx + \int 3x^{3} dx + \int x^{4} dx \)
03

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 \(\frac{1}{2} x^{2} + x^{3} + \frac{3}{4} x^{4} + \frac{1}{5} x^{5}\)
04

Final answer

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

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Most popular questions from this chapter

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. \int_{0}^{2} x^{2} d x, n=4

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1) $$

Quality Control A company manufactures wooden yardsticks. The lengths of the yardsticks are normally distributed with a mean of 36 inches and a standard deviation of \(0.2\) inch. Find the probability that a yardstick is (a) longer than \(35.5\) inches. (b) longer than \(35.9\) inches.

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{1} e^{x^{3}} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \sqrt{1+x^{3}} d x, n=4 $$
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