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Magazine Chapter 25: Problem 30
Integrate each of the given expressions. \(\int 6 x^{2}\left(1-x^{3}\right)^{4 / 3} d x\)
Short Answer
Expert verified
The integral of the expression is \\
\\[-\frac{6}{7}(1-x^{3})^{7/3} + C\\].
Step by step solution
01
Identify the Integration Technique
We have an expression \(6 x^{2}(1-x^{3})^{4/3}\). The term inside the parenthesis, \(1-x^{3}\), suggests we can use a substitution method, where a new variable \(u\) might simplify the integral.
02
Choose the Substitution
Let \(u = 1 - x^{3}\). Then, the derivative of \(u\) with respect to \(x\) is \(-3x^{2}\), which gives \(du = -3x^{2}dx\). To match \(6x^2dx\) in the original integral, multiply both sides of \(du = -3x^{2}dx\) by \(-2\), resulting in \(-2du = 6x^{2}dx\).
03
Substitute and Simplify the Integral
Substitute \(u\) into the integral. Replace \(6x^2dx\) with \(-2du\). The integral becomes: \(\int (1-x^{3})^{4/3}\cdot 6x^2dx = \int u^{4/3} \cdot (-2) du = -2 \int u^{4/3} du\).
04
Integrate with Respect to New Variable
Integrate \(-2 \int u^{4/3} du\). Applying the power rule \(\int u^n du = \frac{u^{n+1}}{n+1} + C\), where \(n = \frac{4}{3}\), gives us:\(-2 \cdot \left(\frac{u^{7/3}}{7/3}\right) + C = -2 \cdot \left(\frac{3}{7}u^{7/3}\right) + C\).Simplify the expression to \(-\frac{6}{7}u^{7/3} + C\).
05
Substitute Back to Original Variable
Replace \(u\) with \(1-x^{3}\) to revert back to the original variable:\(-\frac{6}{7}(1-x^{3})^{7/3} + C\).
06
Final Integration Result
Thus, the final integrated expression is \\[-\frac{6}{7}(1-x^{3})^{7/3} + C\], where \(C\) is the constant of integration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The **Substitution Method** is a powerful tool in integration used to simplify complex integrals. It involves changing variables to make an integral more manageable. In our case, we have the integral \( \int 6 x^{2}(1-x^{3})^{4/3} dx \). The term \( 1-x^3 \) suggests a natural choice for substitution.
- First, choose a substitution. Let \( u = 1 - x^3 \). This substitution simplifies the inside of the power.
- Next, find the derivative. The derivative of \( u \) with respect to \( x \) is \(-3x^2\). Thus, \( du = -3x^2 dx \).
- Adjust the differential. To match \( 6x^2 dx \), multiply the entire differential equation by \(-2\), yielding \( -2du = 6x^2 dx \).
Power Rule in Integration
The **Power Rule in Integration** is a fundamental technique used when integrating functions of the form \( u^n \), where \( n \) is a real number. This method allows us to integrate powers of a variable systematically.To apply the power rule, use the formula:\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]In our scenario:
- We need to integrate \( u^{4/3} \). Thus, using the power rule provides \( \frac{u^{7/3}}{7/3} \) after integration.
- Don鈥檛 forget the constant of integration, \( C \), an essential part of indefinite integrals.
Integral Calculus
**Integral Calculus** is one of the two principal branches of calculus, the other being differential calculus. It involves finding the integral of a function, which is essentially the reverse process of differentiation. Integrals allow us to find areas under curves, total accumulated quantities, and are pivotal in solving real-world problems involving accumulation.
- In our exercise, we are asked to find an unspecified, possibly accumulated, value, which exists as the area under the curve described by \( 6 x^{2}(1-x^{3})^{4/3} \).
- Integration methods like substitution are frequently used in integral calculus to simplify problems into solvable pieces.
Definite and Indefinite Integrals
In **Integral Calculus**, there are two main types of integrals: definite and indefinite integrals. Understanding the distinction is crucial for solving integration problems effectively.
- **Indefinite Integrals** are integrals without upper and lower limits, yielding a general solution plus an arbitrary constant, \( C \). They represent a family of functions: \( \int f(x) \, dx = F(x) + C \).
- **Definite Integrals** have set limits, producing a specific numerical value representing the area under the curve between two points: \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).
