Explanation Result used: If f is a continuous function, ∫ 0 π 2 f ( cos x ) d x = ∫ 0 π 2 f ( sin x ) d x . Theorem used: The Fundamental Theorem of Calculus: Suppose f is continuous on [ a , b ] . Then ∫ a b f ( x ) d x = F ( b ) − F ( a ) , where F is any anti-derivative of f , that is F ′ = f . Calculation: The evaluation of the definite integrals ∫ 0 π 2 cos 2 x d x and ∫ 0 π 2 sin 2 x d x is as follows. Use the result cos 2 x = 1 + cos 2 x 2 and rewrite the integral ∫ 0 π 2 cos 2 x d x as, ∫ 0 π 2 cos 2 x d x = ∫ 0 π 2 ( 1 + cos 2 x 2 ) d x = 1 2 ∫ 0 π 2 ( 1 + cos 2 x ) d x Use the Fundamental Theorem for Calculus, the integral ∫ 0 π 2 cos 2 x d x is calculated as follows. ∫ 0 π 2 cos 2 x d x = 1 2 ∫ 0 π 2 ( 1 + cos 2 x ) d x = 1 2 ∫ 0 π 2 1 d x + 1 2 ∫ 0 π 2 cos 2 x d x = 1 2 <

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Chapter 4.5, Problem 66E

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To evaluate: The integrals ∫0π2cos2x dx and ∫0π2sin2x dx using the Exercise 65.

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Chapter 4.5, Problem 65E

Chapter 4.5, Problem 67E

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Chapter 4 Solutions

Single Variable Calculus

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The integrals ∫ 0 π 2 cos 2 x d x and ∫ 0 π 2 sin 2 x d x using the Exercise 65.

Solution Summary: The author explains the fundamental theorem of Calculus: Suppose f is a continuous function.

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To evaluate: The integrals ∫0π2cos2x dx and ∫0π2sin2x dx using the Exercise 65.

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Chapter 4 Solutions