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Chapter 1: Problem 8

Now evaluate the following integrals. \(\int x(x-1)^{3} d x\)

Short Answer

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

Step by step solution

01

Expand the Polynomial

In order to simplify the polynomial inside the integral, let's expand it. \(x(x-1)^{3}\) becomes \(x(x^{3} - 3x^{2} + 3x - 1)\), which simplifies further to \(x^{4} - 3x^{3} + 3x^{2} - x\).
02

Break up the integral

Now that we have a simplified polynomial, we can break up the integral of the whole polynomial into separate integrals for each term. This gives us \(\int x^{4} dx - \int 3x^{3} dx + \int 3x^{2} dx - \int x dx\).
03

Solve each integral

Each of these integrals can now be solved. Using the power rule for integrals, we get: \(\int x^{4} dx = \frac{1}{5}x^{5}\), \(\int 3x^{3} dx = \frac{3}{4}x^{4}\), \(\int 3x^{2} dx = x^{3}\) and \(\int x dx = \frac{1}{2}x^{2}\)
04

Combine all terms

The last step is to combine all of these integrals: \(\frac{1}{5}x^{5} - \frac{3}{4}x^{4} + x^{3} - \frac{1}{2}x^{2}\). This is the definite answer to the integral computation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expansion
When dealing with integrals involving polynomials, it's often useful to expand any expressions you can. This makes the process of integration much simpler. In our exercise, we had the expression \(x(x-1)^{3}\). To expand it, we distribute \(x\) across the expanded form of \((x-1)^3\), which is \(x^3 - 3x^2 + 3x - 1\).

This results in the polynomial \(x^4 - 3x^3 + 3x^2 - x\). Working with expanded polynomials allows for direct application of the rules of integration. It simplifies complex expressions into a sum of simpler terms.

  • Ensure you correctly distribute terms across the expression.
  • Verify each step to catch any errors early on.
  • Expansion lays the foundation for breaking down the integral later.
Power Rule for Integrals
The power rule is a fundamental concept in integral calculus. It provides a straightforward way to integrate terms of the form \(x^n\). According to the power rule, the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), given that \(n\) is not equal to -1.

For instance, when integrating \(x^4\), we use the power rule to get \(\frac{1}{5}x^5\). Different coefficients in front of the terms, like the \(3\) in \(3x^3\), require adjusting the rule to \(\frac{3}{4}x^4\).

  • Apply the rule to each term individually in a polynomial.
  • Adjust for coefficients outside the \(x^n\) term by multiplying after integration.
  • This rule is essential for efficiently handling polynomial integrals.
Breaking Up Integrals
Breaking up integrals allows for managing complex expressions by addressing each term separately. After expanding the polynomial, \(x^4 - 3x^3 + 3x^2 - x\), we break the integral into parts: \(\int x^4 dx - \int 3x^3 dx + \int 3x^2 dx - \int x dx\).

This method simplifies the problem because you can now apply the power rule to each individual term separately. This makes integration straightforward and avoids confusion from handling all terms at once.

  • Divide the integral into simpler, more manageable pieces.
  • Tackle each component with its own calculation.
  • Ensure to factor in all terms when calculating the overall solution.

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Most popular questions from this chapter

Evaluate \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x\)

\(\int \operatorname{tab}^{6} x \sec ^{2} x d x=\) (A) \(\frac{\tan ^{7} x}{7}+C\) (B) \(\frac{\tan ^{7} x}{7}+\frac{\sec ^{3} x}{3}+C\) (C) \(\frac{\tan ^{7} x \sec ^{3} x}{21}+C\) (D) \(7 \tan ^{7} x+C\)

Now evaluate the following integrals. \(\int \frac{1}{\csc x} d x\)

\(\int_{0}^{\frac{\pi}{4}} \sin x d x+\int_{-\frac{\pi}{4}}^{0} \cos x d x=\) (A) \(-1\) (B) \(0\) (C) \(1\) (D) \(\sqrt{2}\)

\(\int\left(e^{3 \ln x}+e^{3 x}\right) d x=\) (A) \(3+\frac{e^{3 x}}{3}+C\) (B) \(\frac{e^{x^{4}}}{4}+3 e^{3 x}+C\) (C) \(\frac{e^{x^{4}}}{4}+\frac{e^{3 x}}{3}+C\) (D) \(\frac{x^{4}}{4}+\frac{e^{3 x}}{3}+C\)
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