/*! 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 45 Find the indefinite integrals. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 45

Find the indefinite integrals. $$ \int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) d t $$

Short Answer

Expert verified
\( 3\ln |t| - \frac{2}{t} + C \)

Step by step solution

01

Simplify the Integral Expression

The given integral is \( \int\left(\frac{3}{t}-\frac{2}{t^{2}}\right) dt \). Before integrating, simplify the expression by breaking it into separate terms: \( \int \frac{3}{t} dt - \int \frac{2}{t^2} dt \). This will make the integration process easier.
02

Integrate the First Term

Focus on the first term \( \int \frac{3}{t} dt \). Use the formula for the integral of \( \frac{1}{t} \), which is \( \ln |t| \). Multiply by the constant factor, 3, to get \( 3\ln |t| \).
03

Integrate the Second Term

Now, integrate the second term \( \int \frac{2}{t^2} dt \). Rewrite \( \frac{2}{t^2} \) as \( 2t^{-2} \) and apply the power rule for integration: \( \int t^n dt = \frac{t^{n+1}}{n+1} + C \). This gives \( -2t^{-1} = -\frac{2}{t} \).
04

Combine the Results and Include the Constant of Integration

Combine the results from Step 2 and Step 3. The overall integral becomes \( 3\ln |t| - \frac{2}{t} + C \), where \( C \) is the constant of integration included in indefinite integrals.

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 Techniques
Integration is a fundamental concept in calculus used to find areas, volumes, central points, and many useful things. When dealing with indefinite integrals, our goal is to find a function whose derivative gives us the integrand. There are several techniques to approach integrations, such as substitution, integration by parts, and partial fractions, but here we focus on simpler techniques.
  • Simplifying the expression first can make the integration process more manageable. For example, splitting a complex integral into simpler parts can often lead to easier integration.
  • Understanding how to handle each part individually means we can apply specific rules, like the power rule or logarithmic rule, to different components.
In the given exercise, breaking down the integral into two separate terms made it possible to address each with its appropriate technique, transforming the problem into manageable pieces.
Power Rule
The power rule is a straightforward and essential tool used in integration (and differentiation). It applies to polynomials and involves the process of raising the power of a term by one and then dividing by the new power.When you encounter an integral of the form \( \int x^n \, dx \), use the power rule: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \( n eq -1 \). This elegant rule simplifies the integration of terms with powers.
  • The exercise involved the term \( \frac{2}{t^2} \), which can be rewritten as \( 2t^{-2} \). Applying the power rule gave \( -\frac{2}{t} \) because we first increase the exponent \( n \) from \(-2\) to \(-1\) and then divide by the new exponent.
  • This process shows how fundamental rules can make tough integrals approachable, step by step.
By recognizing this, indefinite integrals become solvable in a series of straightforward applications.
Constant of Integration
When solving indefinite integrals, the constant of integration, denoted as \( C \), plays a crucial role. It accounts for the family of all antiderivatives, since integration is essentially the reverse process of differentiation.Differentiating a function removes any constant value because the derivative of a constant is zero. Therefore, when integrating, we must include \( C \) to signify any constant that might have been present in the original function before differentiation.
  • Without this constant, we鈥檇 miss representing the complete set of functions that satisfy the integrand when differentiated.
  • In the solution from the exercise, after finding the antiderivative forms like \( 3\ln |t| - \frac{2}{t} \), the addition of \( C \) reflects that any solution differs from another by a constant amount.
This constant ensures that our solution is as general as possible, covering all potential original functions.

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

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{e^{t}}{e^{t}+1} d t $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1} 2 e^{x} d x $$

Graph \(y=1 / x^{2}\) and \(y=1 / x^{3}\) on the same axes. Which do you think is larger: \(\int_{1}^{\infty} 1 / x^{2} d x\) or \(\int_{1}^{\infty} 1 / x^{3} d x\) ? Why?

Use integration by parts twice to evaluate the integral. $$ \int t^{2} e^{5 t} d t $$

Derive the formula (called a reduction formula): $$ \int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied 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.