/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Calculate the indefinite integra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 9

Calculate the indefinite integral. $$ \int(x+1)^{2} d x $$

Short Answer

Expert verified
The indefinite integral is \( \frac{x^3}{3} + x^2 + x + C \).

Step by step solution

01

Expand the Integrand

To solve the integral, start by expanding the expression \(x + 1\)^2. This gives us: \\[(x + 1)^2 = x^2 + 2x + 1\]
02

Integrate Each Term Separately

Now that we have \(x^2 + 2x + 1\), we can integrate each term separately: \- Integrate \(x^2\): \int x^2 \, dx = \frac{x^3}{3}\- Integrate \(2x\): \int 2x \, dx = x^2\- Integrate \(1\): \int 1 \, dx = x\.\Combine these results to get the integrated expression.
03

Combine the Results

Combine the integrals of each term: - \frac{x^3}{3} + x^2 + x\Don't forget to add the constant of integration, \C\. Thus, the indefinite integral is \[\frac{x^3}{3} + x^2 + x + C\].
04

Verify the Results

To verify, differentiate the result \[\frac{x^3}{3} + x^2 + x + C\] with respect to \(x\) and check that it matches the original expression: - \frac{d}{dx}\left(\frac{x^3}{3}\right) = x^2\ - \frac{d}{dx}(x^2) = 2x\ - \frac{d}{dx}(x) = 1\The result \(x^2 + 2x + 1\) matches the expanded form of the original integrand, confirming the solution is correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integration by Substitution
Integration by substitution is a powerful technique that aims to simplify integrals by making a smart substitution. It's like untangling a knot by finding a clever thread to pull. You replace a part of the integrand with a new variable, which often simplifies the integral into a form that's easier to work with.

Here鈥檚 how it works:
  • Identify a part of the integrand that can be substituted with a new variable, say \( u \).
  • Express \( dx \) in terms of \( du \) by differentiating your substitution.
  • Rewrite the entire integral in terms of \( u \) and \( du \).
  • Integrate with respect to \( u \).
  • Substitute back into the original variable.
Substitution is especially useful when dealing with integrals involving composite functions or expressions that resemble the derivative of another expression.
Integration Techniques
Integration is a crucial tool in calculus, similar to finding the area under a curve. However, not all functions can be integrated directly, which is why different techniques exist to tackle various types of integrals.

Some common integration techniques include:
  • Basic Integrals: These involve straightforward functions like \( x^n \), where you integrate directly using power rules.
  • Integration by Parts: Useful when an integrand is the product of two functions, relying on the product rule of differentiation.
  • Partial Fraction Decomposition: Helps integrate rational functions by breaking them into simpler fractions.
  • Trigonometric Substitution: Used for integrals involving expressions like \( \sqrt{a^2 - x^2} \).
Knowing when and how to use each technique is key to solving complex integrals efficiently.
Expanding Expressions
Expanding an expression can significantly simplify the process of integration, as seen in our example.
  • Firstly, understand the expression: What does it represent?
  • Next, rewrite it in a form that makes it easier to handle. For example, \((x+1)^2\) expands into \(x^2 + 2x + 1\).
  • Once expanded, you integrate each term separately.
In many cases, expansion helps remove the complexity of dealing with powers or composite expressions directly, thus allowing for simple integration term by term. Always remember, after integrating, combine all terms together and add the constant of integration \( C \). This complete expression represents the indefinite integral you seek.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In each of Exercises 54-60, determine each point \(c,\) where the given function \(f\) satisfies \(f^{\prime}(c)=0\). At each such point, use the First Derivative Test to determine whether \(f\) has a local maximum, a local minimum, or neither. $$ f(x)=\tanh ^{-1}(\sqrt{x})-2 \sqrt{x} $$

Calculate the indefinite integral. $$ \int x^{1 / 2}(x+1) d x $$

The initial temperature \(T(0)\) of an object is \(50^{\circ} \mathrm{C}\). If the object is cooling at the rate $$T^{\prime}(t)=-0.2 e^{-0.02 t}$$ when measured in degrees centigrade per second, then what is the limit of its temperature as \(t\) tends to infinity?

Compute \(F(c)\) from the given information. $$ F^{\prime}(x)=2 x+3, F(1)=2, c=-1 $$

Use trigonometric identities to compute the indefinite integrals. $$ \int \sin (x) \cos (x) d x $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.