/*! 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 1 (a) \(\int\left(1-3 x^{2}\right)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

(a) \(\int\left(1-3 x^{2}\right) d x\) (b) \(\int x^{-4} d x\) (c) \(\int\left(1-x^{2}\right)^{2} d x\)

Short Answer

Expert verified
(a) x - x^3 + C; (b) -\frac{1}{3x^3} + C; (c) x - \frac{2}{3}x^3 + \frac{1}{5}x^5 + C

Step by step solution

01

- Solve part (a) by integrating

To solve the integral \(\int\left(1-3 x^{2}\right) d x\), integrate each term separately: \[ \int 1 \, dx - \int 3x^2 \, dx = x - x^3 \] Add the constant of integration, C: \[ x - x^3 + C \]
02

- Solve part (b) by integrating

To solve the integral \(\int x^{-4} dx\), use the formula for integrating power functions \(\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\). Here, n = -4: \[ \int x^{-4} \, dx = \frac{x^{-3}}{-3} = -\frac{1}{3}x^{-3} + C \]
03

- Simplify the result for part (b)

Rewrite the result using positive exponents: \[ -\frac{1}{3}x^{-3} = -\frac{1}{3x^3} + C \]
04

- Expand the integrand for part (c)

To solve \(\int\left(1-x^{2}\right)^{2} dx\), expand the integrand: \[ \left(1-x^{2}\right)^{2} = 1 - 2x^2 + x^4 \]
05

- Integrate each term for part (c)

Integrate each term separately: \[ \int\left(1 - 2x^2 + x^4\right) dx = \int 1 \, dx - 2\int x^2 \, dx + \int x^4 \, dx \] This simplifies to: \[ x - \frac{2}{3}x^3 + \frac{1}{5}x^5 + C \]

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.

Definite Integrals
Unlike indefinite integrals, definite integrals have upper and lower limits. When calculating a definite integral, you essentially find the area under the curve for a specific interval. This interval is represented by the limits of integration. For example, in the notation \(\int_{a}^{b} f(x) \, dx \), \a\ and \b\ are the limits of integration.

Here are the steps to solve a definite integral:
  • Find the indefinite integral first.
  • Evaluate the resulting function at the upper limit.
  • Subtract the value of the function at the lower limit from the value at the upper limit.
It's important to understand that the definite integral of a function can be used to find areas, volumes, and other physical properties in various fields such as physics and engineering.
Indefinite Integrals
Indefinite integrals, or antiderivatives, represent a family of functions and include a constant of integration, \C\. When finding the indefinite integral, you're essentially reversing the process of differentiation. Let's break down how to solve part (a) of our exercise:

To solve \(\int \left(1-3 x^{2}\right) dx\), integrate each term separately:
  • The integral of \1\ with respect to \x\ is \x\.
  • The integral of \-3x^2\ is \-x^3\ (using the power rule).
So, we get: \x - x^3 + C\

This process basically undoes differentiation, allowing us to find the original function that was differentiated.
Power Functions
A power function has the form \f(x) = x^n\, where \ is any real number. When integrating power functions, you use the formula \ \int x^n \,dx = \frac{x^{n+1}}{n+1} + C \. Let鈥檚 see this in action with part (b) and part (c) of our exercise:

For part (b), \ \int x^{-4} \, dx \ involves applying this formula with \ = -4\:
\ \int x^{-4} \, dx = \frac{x^{-3}}{-3} = -\frac{1}{3}x^{-3} + C \

For part (c), \( \int\left(1-x^{2}\right)^{2} dx \) requires first expanding \( (1 - x^2)^2 \) into \(1 - 2x^2 + x^4\). Then we integrate each term:
  • \ \int 1 \, dx = x \
  • \ \int -2x^2 \, dx = - \frac{2}{3}x^3 \
  • \ \int x^4 \, dx = \frac{1}{5}x^5 \

Combining these, we get: \ x - \frac{2}{3}x^3 + \frac{1}{5}x^5 + C \

Understanding how to integrate power functions is essential for solving a wide range of calculus problems.

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

(a) Sketch the domain of integration and compute the integral \(\int_{0}^{1}\left(\int_{x^{2}}^{x}\left(x^{2}+x y\right) d y\right) d x\). (b) Change the order of integration and verify that you obtain the same result as in (a).

Consider the function \(f(x, y)=\frac{2 k}{(x+y+1)^{3}}\) (k constant). Let \(R\) be the rectangle \(R=[0, a] \times[0,1]\), where \(a>0\) is a constant. Determine the value \(k_{a}\) of \(k\) such that \(\iint_{R} f(x, y) d x d y=1 .\) Show that \(k_{a}>2\) for all \(a>0\)

(a) \(\int_{0}^{1} \frac{4 x^{3}}{\sqrt{4-x^{2}}} d x\) (b) \(\int_{1}^{8} \frac{1}{3+\sqrt{t+8}} d t\) (c) \(\int_{1}^{e^{2}} \sqrt{x} \ln x d x\)

$$ \text { Compute the double integral } I=\int_{0}^{2}\left(\int_{-2}^{1}\left(x^{2} y^{3}-(y+1)^{2}\right) d y\right) d x \text {. } $$

(a) \(\int_{0}^{10}\left(10 t^{2}-t^{3}\right) d t\) (b) \(\int_{0}^{10} 4 t e^{-2 t} d t\) (c) \(\int_{0}^{10} \frac{10 t^{2}-t^{3}}{t+1} d t\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

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.