   Chapter 8.5, Problem 31E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Integrating a Trigonometric Function In Exercises 1-32, find the indefinite integral. ∫ ( sin x + cos x ) 2 d x

To determine

To calculate: The indefinite integral, (sinx+cosx)2dx.

Explanation

Given Information:

The provided integral is (sinx+cosx)2dx.

Formula used:

Write the formula for integral of sinu.

sinudu=cosu+C

Write the formula of integral of du.

du=u+C

Write the formulae for trigonometrical identities.

sin2u+cos2u=1,

And,

2sinucosu=2sin2u

Calculation:

Consider the provided integral,

(sinx+cosx)2dx=(sin2x+cos2x+2sinxcosx)dx=(1+2sinxcosx)dx=(1+sin2x)dx

Let u=2x.

Differentiate the above considered function.

du=2dx

The provided integral is,

(sinx+cosx)2dx

Use the formulae of trigonometrical identities to simplify the provided integral.

(sinx+cosx)2dx=dx+sin2xdx

Multiply and divide by 2

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