# Evaluating Indefinite integrals

• n77ler
In summary, an indefinite integral is the reverse process of differentiation and represents a family of functions whose derivative is equal to the given function. To evaluate an indefinite integral, integration techniques such as substitution, integration by parts, or partial fractions can be used. The difference between definite and indefinite integrals lies in their limits and applications, with definite integrals giving a numerical value and being used to find the area under a curve, while indefinite integrals represent a family of functions and are used to find the original function. Some common mistakes in evaluating indefinite integrals include forgetting the constant of integration, not applying the chain rule correctly, and not simplifying the integrand. Indefinite integrals are applied in various real-life situations, such as modeling and
n77ler

## Homework Statement

Evaluate

$$\oint x^2 (1-x^3)^6 dx$$

## The Attempt at a Solution

let u= 1-x^3
du= -3x^2
-1/3 du= x^2 dx
-1/3 $$\oint (u)^6$$
= -1/3 (u^7/7)
= -1/21 (1-x^3)^7 + C

Is this done correct? I think I followed all the right steps but there is something about it that has me wondering.

the only problem I see is that circle on your integral symbol!

What do you get if you differentiate -(1/21)(1-x3)7+ C?

Yeah, the circle isn't supposed to be there I couldn't find a normal Integration sign.

## 1. What is an indefinite integral?

An indefinite integral is the reverse process of differentiation. It represents a family of functions whose derivative is equal to the given function. It is denoted by the symbol ∫.

## 2. How do you evaluate an indefinite integral?

To evaluate an indefinite integral, you need to use integration techniques such as substitution, integration by parts, or partial fractions. You can also use tables of integrals or online calculators to obtain the result.

## 3. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and gives a numerical value, while an indefinite integral has no limits and represents a family of functions. A definite integral can be used to find the area under a curve, while an indefinite integral is used to find the original function.

## 4. What are some common mistakes in evaluating indefinite integrals?

Some common mistakes include forgetting to add the constant of integration, not applying the chain rule correctly, and not simplifying the integrand before integrating. It is also important to check for any errors in arithmetic or algebraic manipulation.

## 5. How can indefinite integrals be applied in real-life situations?

Indefinite integrals are used in various fields of science and engineering, such as physics, economics, and computer science. They can be used to model and analyze real-life situations, such as predicting the growth of a population or calculating the work done by a force.

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