Chapter 13: Chapter 13

Problem 10

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Evaluate the given integral using the substitution (or method) indicated. $$ \int(x-1)^{2} e^{(x-1)^{3}} d x ; u=(x-1)^{3} $$

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The short version of the answer is: $$ \int (x-1)^2 e^{(x-1)^3}\, dx = \frac{1}{3} e^{(x-1)^3} + C $$

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identify the given integral and substitution

We are given the integral to evaluate: $$ \int (x-1)^2 e^{(x-1)^3} dx $$ The substitution function is given by: $$ u = (x-1)^3 $$

Step 2: Find the derivative of the substitution function

Calculate $du/dx$ of the substitution $u = (x-1)^3$: $$ \frac{du}{dx} = 3(x-1)^2 $$ Now, calculate $dx$ by rearranging this equation: $$ dx = \frac{du}{3(x-1)^2} $$

Step 3: Rewrite the integral in terms of u

Using the substitution function, we can see that: $$ (x-1)^2 = \sqrt[3]{u} $$ Replace $(x-1)^2$ with $\sqrt[3]{u}$ and $dx$ with $\frac{du}{3\sqrt[3]{u^2}}$ in the integral: $$ \int (x-1)^2 e^{(x-1)^3}\, dx = \int \sqrt[3]{u} e^{u} \cdot \frac{du}{3\sqrt[3]{u^2}} $$

Step 4: Simplify the integral

Notice that we can simplify the expression in the integral: $$ \sqrt[3]{u} \cdot \frac{1}{3\sqrt[3]{u^2}} = \frac{1}{3\sqrt[3]{u}} $$ Now, we can rewrite the integral as: $$ \int \frac{1}{3\sqrt[3]{u}} e^u\, du $$

Step 5: Evaluate the new integral

We can pull the constant 1/3 out of the integral and simply integrate $e^u$ with respect to $u$: $$ \frac{1}{3} \int \frac{1}{\sqrt[3]{u}} e^u\, du = \frac{1}{3} \int e^u\, du $$ Now, integrate: $$ \frac{1}{3} \int e^u\, du = \frac{1}{3} e^u + C $$ Where $C$ is the constant of integration.

Step 6: Substitute back the original variable x

Now, we substitute back our original substitution $u = (x-1)^3$ to find the resulting function of $x$: $$ \frac{1}{3} e^u + C = \frac{1}{3} e^{(x-1)^3} + C $$ So, the final result is: $$ \int (x-1)^2 e^{(x-1)^3}\, dx = \frac{1}{3} e^{(x-1)^3} + C $$

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Chapter 13: Chapter 13

Evaluate the given integral using the substitution (or method) indicated. $$ \int(x-1)^{2} e^{(x-1)^{3}} d x ; u=(x-1)^{3} $$

Short Answer

Step by step solution

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Step 1: Identify the given integral and substitution

Step 2: Find the derivative of the substitution function

Step 3: Rewrite the integral in terms of u

Step 4: Simplify the integral

Step 5: Evaluate the new integral

Step 6: Substitute back the original variable x

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Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 14

Chapter 15

Chapter 16

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