Chapter 16: Chapter 16

Problem 12

Evaluate the integrals. \(\int(x+1)\left[\cos \left(x^{2}+2 x\right)+\left(x^{2}+2 x\right)\right] d x\)

Short Answer

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The short answer for the integral \(\int(x+1)\left[\cos \left(x^{2}+2 x\right)+\left(x^{2}+2 x\right)\right] d x\) is: \(\frac{1}{2}\sin{x^2+2x} + \frac{1}{4}x^4 + x^3 + x^2 + C\)

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Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Separate the integral

We have: \(\int(x+1)\left[\cos \left(x^{2}+2 x\right)+\left(x^{2}+2 x\right)\right] d x = \int(x+1)\cos \left(x^{2}+2 x\right) dx + \int(x+1)\left(x^{2}+2 x\right) dx\)

Step 2: Evaluate the integral of the first part

Now, to find the integral of the first part: \(\int(x+1)\cos \left(x^{2}+2 x\right) dx\) Here, we have to use the substitution method: Let \(u = x^2 + 2x\), then \(\frac{d u}{d x} = 2x + 2\). Hence, the integral becomes \(\int \frac{x+1}{2x+2}\cos{(u)} d u\) Let's simplify: \(\int \frac{x+1}{2x+2}\cos{(u)} d u = \frac{1}{2}\int\cos{(u)} d u\) Now, integrate the expression: \(\frac{1}{2}\int\cos{(u)} d u = \frac{1}{2}\sin{(u)} + C_1\) Let's substitute back the value of u: \(\frac{1}{2}\sin{x^2+2x} + C_1\)

Step 3: Evaluate the integral of the second part

Now, to find the integral of the second part: \(\int(x+1)\left(x^{2}+2 x\right) dx\) Integration using polynomial rules: \(\int(x+1)\left(x^{2}+2 x\right) dx = \int x^3 + 3x^2 + 2x dx\) Now we integrate each term: \(\frac{1}{4}x^4 + x^3 + x^2 + C_2\)

Step 4: Combine the results

Now, we combine the results of the two integrals: \[\frac{1}{2}\sin{x^2+2x} + \frac{1}{4}x^4 + x^3 + x^2 + C\] where \(C = C_1 + C_2\) is the constant of integration. So, the final result is: \(\int(x+1)\left[\cos \left(x^{2}+2 x\right)+\left(x^{2}+2 x\right)\right] d x = \frac{1}{2}\sin{x^2+2x} + \frac{1}{4}x^4 + x^3 + x^2 + C\)

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Chapter 16: Chapter 16

Evaluate the integrals. \(\int(x+1)\left[\cos \left(x^{2}+2 x\right)+\left(x^{2}+2 x\right)\right] d x\)

Short Answer

Step by step solution

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Step 1: Separate the integral

Step 2: Evaluate the integral of the first part

Step 3: Evaluate the integral of the second part

Step 4: Combine the results

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More chapters from the book ‘Finite Mathematics and Applied Calculus’

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 13

Chapter 14

Chapter 15

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