/*! 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 19 Find the most general antideriva... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 19

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. $$\int\left(3 t^{2}+\frac{t}{2}\right) d t$$

Short Answer

Expert verified
The most general antiderivative is \( t^3 + \frac{t^2}{4} + C \).

Step by step solution

01

Identify Separate Terms

The integrand consists of two terms: \(3t^2\) and \(\frac{t}{2}\). We can integrate each term separately according to the properties of indefinite integrals.
02

Integrate the First Term

Apply the power rule for integration to the first term: \( \int 3t^2 \, dt \). According to the power rule, \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \). Here, \( n = 2 \), so we have \( \frac{3t^{3}}{3} = t^3 \).
03

Integrate the Second Term

For the second term, apply the power rule again: \( \int \frac{t}{2} \, dt \). Treat \( \frac{1}{2} \) as a constant, giving us \( \frac{1}{2} \int t \, dt \). By the power rule, \( \frac{1}{2} \cdot \frac{t^2}{2} = \frac{t^2}{4} \).
04

Combine the Results

Combine the antiderivatives found in Steps 2 and 3: \( t^3 + \frac{t^2}{4} + C \), where \( C \) is the constant of integration.
05

Verify by Differentiation

Differentiate the result \( t^3 + \frac{t^2}{4} + C \) to verify correctness. The derivative is \( 3t^2 + \frac{t}{2} \), which matches our original integrand. Therefore, the integration 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.

Power Rule for Integration
The power rule for integration is a fundamental concept in calculus. It's a method used to find the antiderivative of polynomial expressions. The rule states:
  • If you have an expression in the form of \( t^n \), its integral is \( \frac{t^{n+1}}{n+1} \), where \( n eq -1 \).
  • Always add a constant of integration, \( C \), at the end.
This rule transforms an algebraic expression into its antiderivative by increasing the exponent by one and dividing by the new exponent.
For example, to integrate \( 3t^2 \), we apply the power rule:
  • Increase the exponent of \( t^2 \) to 3, leading to \( t^3 \).
  • Then divide by the new exponent. Therefore, the integral of \( 3t^2 \) is \( \frac{3t^3}{3} = t^3 \).
The power rule is a quick method to solve polynomial integrals and is widely used in solving calculus problems.
Indefinite Integral
An indefinite integral is a function that represents the collection of all antiderivatives of a given expression. In essence, it is the general solution to a differential equation. Unlike definite integrals, which provide a numerical result, an indefinite integral takes the form of a function.
  • It is represented with the integral sign \( \int \) followed by the function you want to integrate, ending with \( + C \).
  • The \( + C \) represents an arbitrary constant, as there are infinitely many antiderivatives differing by a constant.
For example, the indefinite integral \( \int (3t^2 + \frac{t}{2}) \, dt \) requires finding antiderivatives for each term:
  • The first term, \( 3t^2 \), integrates to \( t^3 \).
  • The second term, \( \frac{t}{2} \), integrates to \( \frac{t^2}{4} \).
Combining both gives us the indefinite integral: \( t^3 + \frac{t^2}{4} + C \). This is the most general form of the antiderivative for the given expression.
Constant of Integration
The constant of integration, denoted as \( C \), is an essential part of indefinite integrals. When we integrate a function, there are infinitely many solutions because the derivative of a constant is zero.
  • Without \( C \), the solution to the indefinite integral would be incomplete.
  • The constant represents all possible vertical shifts of the antiderivative graph.
Therefore, whenever you perform integration, it's crucial to include \( + C \) in your answer.
In the example of \( \int (3t^2 + \frac{t}{2}) \, dt \), the solution is \( t^3 + \frac{t^2}{4} + C \).
  • If you differentiate this function, \( \frac{d}{dt} (t^3 + \frac{t^2}{4} + C) \), you return to the original function \( 3t^2 + \frac{t}{2} \) because the derivative of \( C \) is zero.
This demonstrates that the constant \( C \) doesn't affect the validity of the solution but ensures its completeness.

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 most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\begin{aligned} &\int\left(1+\tan ^{2} \theta\right) d \theta\\\ &\text { (Hint: }\left.1+\tan ^{2} \theta=\sec ^{2} \theta\right) \end{aligned}$$

Solve the initial value problems. $$\frac{d^{2} y}{d x^{2}}=0 ; \quad y^{\prime}(0)=2, \quad y(0)=0$$

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. $$\int\left(\frac{1}{5}-\frac{2}{x^{3}}+2 x\right) d x$$

Show that if \(a, b, c,\) and \(d\) are positive integers, then $$\frac{\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)\left(d^{2}+1\right)}{a b c d} \geq 16$$

Solve the initial value problems. $$\frac{d^{2} r}{d t^{2}}=\frac{2}{t^{3}} ;\left.\quad \frac{d r}{d t}\right|_{t=1}=1, \quad r(1)=1$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.