Explanation Given: ∫ 2 sin x cos x d x Concept Used: ∫ a f ( x ) d x = a ∫ f ( x ) d x Trigonometric property: sin 2 x = 2 sin x cos x d d x cos x = − sin x ∫ sin x d x = − sin x + C , where C is an arbitrary constant. Calculation: To solve the integrals ∫ 2 sin x cos x d x First simplify it using trigonometric property, sin 2 x = 2 sin x cos x , as ∫ 2 sin x cos x d x = ∫ sin ( 2 x ) d x Now substitute 2 x = u ⇒ d ( 2 x ) = d u ⇒ 2 d x = d u ⇒ d x = d u 2 Thus given integral changes as ∫ sin ( 2 x ) d x = ∫ sin u d u 2 = 1 2 ∫ sin u d u Now use fundamental formulae of integrals to solve it further, as ∫ 2 sin x cos x d x = 1 2 ∫ sin u d u = 1 2 ( − cos u ) + C = − 1 2 cos u + C Where C is an arbitrary constant

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Question

Chapter 4.4, Problem 44S

To determine

To solve the integrals

∫2sinxcosxdx

And also check the answer by differentiating.

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Chapter 4.4, Problem 43S

Chapter 4.4, Problem 45S

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