## Calculus, 10th Edition (Anton)

Published by Wiley

# Chapter 4 - Integration - 4.9 Evaluating Definite Integrals By Substitution - Exercises Set 4.9 - Page 340: 27

#### Answer

$$1$$

#### Work Step by Step

\eqalign{ & \int_0^{\pi /4} {4\sin x\cos x} dx \cr & u = \sin x,{\text{ then }}du = \cos xdx \cr & {\text{With this substitution}}{\text{,}} \cr & {\text{if }}x = \pi /4,{\text{ }}u = sin\left( {\pi /4} \right) = \frac{{\sqrt 2 }}{2} \cr & {\text{if }}x = 0,{\text{ }}u = \sin \left( 0 \right) = 0 \cr & {\text{so}} \cr & \int_0^{\pi /4} {4\sin x\cos x} dx = \int_0^{\sqrt 2 /2} {4udu} \cr & = 4\int_0^{\sqrt 2 /2} {udu} \cr & {\text{find the antiderivative }} \cr & = 4\left( {\frac{{{u^2}}}{2}} \right)_0^{\sqrt 2 /2} \cr & {\text{part 1 of fundamental theorem of calculus}} \cr & = 2\left( {{{\left( {\frac{{\sqrt 2 }}{2}} \right)}^2} - {{\left( 0 \right)}^2}} \right) \cr & = 2\left( {\frac{1}{2}} \right) \cr & = 1 \cr}

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