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Chapter 4: Problem 22

In Exercises \(17-56,\) find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(1-x^{2}-3 x^{5}\right) d x$$

Short Answer

Expert verified
The most general antiderivative is \( x - \frac{x^3}{3} - \frac{x^6}{2} + C \).

Step by step solution

01

Identify the Structure of the Integral

The given integral is \( \int (1 - x^2 - 3x^5) \, dx \). Recognize that the integral can be split into the sum of the integrals of each term: \( \int 1 \, dx - \int x^2 \, dx - 3 \int x^5 \, dx \). We will integrate each term individually.
02

Integrate the Constant Term

The integral of a constant \( c \) is \( cx \). Thus, the integral of \( 1 \) is \( x \). So, \( \int 1 \, dx = x \).
03

Integrate the Power of x

To integrate a power of \( x \), use the formula \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \). For \( \int x^2 \, dx \), \( n = 2 \), so the integration is \( \frac{x^{3}}{3} \).
04

Integrate with a Coefficient

The integration of \(-3x^5\) requires applying the same power rule, considering the coefficient: \( -3 \int x^5 \, dx = -3 \cdot \frac{x^{6}}{6} \). Simplifying gives \( -\frac{x^{6}}{2} \).
05

Sum the Integrals

Combine the integrals from previous steps: \( x - \frac{x^{3}}{3} - \frac{x^{6}}{2} \). This is the antiderivative of the original function.
06

Include the Constant of Integration

Since indefinite integrals include a constant of integration, add \( C \) to the result: \( x - \frac{x^{3}}{3} - \frac{x^{6}}{2} + C \).
07

Verify by Differentiation

Differentiate the antiderivative \( x - \frac{x^{3}}{3} - \frac{x^{6}}{2} + C \) to ensure it returns to the original function. The derivative is \( 1 - x^2 - 3x^5 \), matching the integrand, confirming the solution is correct.

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Key Concepts

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

Antiderivative
An antiderivative is like stepping back to see where a derivative came from. In simple terms, if you have a function's derivative, the antiderivative helps you find the original function before differentiation. Imagine working backward from knowing how something changes to understanding what it is.
  • Antiderivatives are also known as indefinite integrals.
  • It is important to integrate each term in a function separately to find the antiderivative, as shown in the example with the integral \( \int (1 - x^2 - 3x^5) \, dx \).
  • The process of finding an antiderivative essentially reverses the operation of differentiation.
When you find an antiderivative, like in the example, you are looking for a function whose derivative is the function you started with. Here, integrating each part led to the function \( x - \frac{x^{3}}{3} - \frac{x^{6}}{2} \).
Integration by Parts
Integration by parts is like an advanced tool in our integration toolbox. It helps us integrate products of functions, especially when a simple integration isn't enough. The process is based on the product rule for derivatives but works backward, using the formula:\[\int u \, dv = uv - \int v \, du\]However, in the given exercise, we didn't need to apply integration by parts because each term integrated is a straightforward power of \(x\) or a constant. Despite that, it's always helpful to remember this method because sometimes, integration by parts can be the key to solving more complex problems where functions are multiplied together.Understanding when and how to use this method can save a lot of headaches, especially when tackling more complicated integrals.
Constant of Integration
In calculus, whenever you determine the indefinite integral of a function, it's crucial to include a "\(+ C\)." This \(+ C\) is known as the constant of integration. It's essential because differentiation, in the opposite process, erases constant terms.
  • The constant represents any constant value that could have been present in the original function before differentiation.
  • When you are adding this constant, you are acknowledging that many functions might have the same derivative.
Take the solution from our exercise: \( x - \frac{x^{3}}{3} - \frac{x^{6}}{2} + C \). The \(+ C\) means there is not just one function, but a whole family of functions, all differing by some constant, that could be the original before finding their derivative.

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