/*! 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 2 Find all antiderivatives of each... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 2

Find all antiderivatives of each following function: $$f(x)=9 x^{8}$$

Short Answer

Expert verified
The antiderivative of \(f(x) = 9x^8\) is \( x^9 + C \).

Step by step solution

01

Identify the function to be integrated

The function given for finding the antiderivative is \(f(x) = 9x^8\).
02

Use the power rule for integration

The power rule for integration states that the antiderivative of \(ax^n\) is \( \frac{a}{n+1} x^{n+1} + C \), where \(C\) is the constant of integration. Here, \(a = 9\) and \(n = 8\).
03

Apply the power rule

Using the power rule, we integrate \(9x^8\):\[ \frac{9}{8+1} x^{8+1} + C = \frac{9}{9} x^9 + C = x^9 + C.\]
04

Write the general antiderivative

Thus, the general antiderivative of the function \(f(x) = 9x^8\) is \( x^9 + C \).

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.

power rule for integration
To find the antiderivative of a function like the one given, we use the power rule for integration. The power rule is a fundamental principle in calculus. It helps us integrate functions of the form \(ax^n\). Here is how it works:

The formula for the power rule is \(\frac{a}{n+1} x^{n+1} + C\), where:

  • \

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

Use a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with selected points as specified. \(f(x)=e^{-x} ; 2 \leq x \leq 3, n=5\), right endpoints

Find the area of the region between the curve and the \(x\) -axis. \(f(x)=x^{2}-2 x-3\), from 0 to 2 .

A conical-shaped tank is being drained. The height of the water level in the tank is decreasing at the rate \(h^{\prime}(t)=-\frac{t}{2}\) inches per minute. Find the decrease in the depth of the water in the tank during the time interval \(2 \leq t \leq 4\).

Find the area of the region between the curves. \(y=x^{2}-6 x+12\) and \(y=1\) from \(x=0\) to \(x=4\)

Find the area of the region between the curves. \(y=e^{2 x}\) and \(y=1-x\) from \(x=0\) to \(x=1\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.