Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Evaluate the integral :
\(\int {\frac{{{x^4}}}{{x - 1}}dx} \)

First, divide \({x^4}\) by \(x - 1\), and then proceed further.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Given Data

\(\int {\frac{{{x^4}}}{{x - 1}}dx} = \int {{x^3} + {x^2} + x + 1 + \frac{1}{{x - 1}}dx} \)

Step 2: Splitting the single integral into multiple integrals.

\( = \int {{x^3} dx + \int {{x^2}} dx + \int x dx + \int 1 dx + \int {\frac{1}{{x - 1}}} dx} \)

Step 3: Solving each integral.

\(\begin{array}{l} = \frac{{{x^4}}}{4} + \frac{{{x^3}}}{3} + \frac{{{x^2}}}{2} + x + \int {\frac{1}{{x - 1}}dx} \\\left( {Formula\,used:\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}}} } \right)\end{array}\)

Step 4: Calculation of \(\int {\frac{1}{{x - 1}}dx} \).

Let \(x - 1\) be \(x\)

\(x - 1 = t\)

\(t = x - 1\)

\(\frac{{dt}}{{dx}} = 1 \Rightarrow dt = dx\)

Therefore, by substitution,

\(\begin{array}{l}\int {\frac{1}{{x - 1}}dx} = \int {\frac{1}{t}dt} \\ = log\left| t \right|\\ = log\left| {x - 1} \right|\end{array}\)

\(\int {\frac{{{x^4}}}{{x - 1}}} = \frac{{{x^4}}}{4} + \frac{{{x^3}}}{3} + \frac{{{x^2}}}{2} + x + log\left| {x - 1} \right| + c\)

Hence, the value of the integral \(\int {\frac{{{x^4}}}{{x - 1}}} dx\) is \(\frac{{{x^4}}}{4} + \frac{{{x^3}}}{3} + \frac{{{x^2}}}{2} + x + log\left| {x - 1} \right| + c\)

Find Study Materials for

Create Study Materials

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integral :
\(\int {\frac{{{x^4}}}{{x - 1}}dx} \)

Step by Step Solution

Step 1:Given Data

Step 2: Splitting the single integral into multiple integrals.

Step 3: Solving each integral.

Step 4: Calculation of \(\int {\frac{1}{{x - 1}}dx} \).

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Evaluate the integral : \(\int {\frac{{{x^4}}}{{x - 1}}dx} \)

Step by Step Solution

Step 1:Given Data

Step 2: Splitting the single integral into multiple integrals.

Step 3: Solving each integral.

Step 4: Calculation of \(\int {\frac{1}{{x - 1}}dx} \).

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Evaluate the integral :
\(\int {\frac{{{x^4}}}{{x - 1}}dx} \)