Explanation Given information: The indefinite integral, ∫ cos 3 ( π θ ) sin 4 ( π θ ) d θ Concept used: cos 2 x = 1 − sin 2 x ∫ cos x d x = sin x Calculation: To evaluate the indefinite integral we use the u-substitution. So on substituting, π θ = u ⇒ π d θ = d u ⇒ d θ = 1 π d u , the given integral becomes ∫ cos 3 ( π θ ) sin 4 ( π θ ) d θ = ∫ cos 3 u sin 4 u ⋅ 1 π d u = 1 π ∫ cos 3 u sin 4 u d u To simplify it further, we use the Pythagorean identity. So we get ∫ cos 3 ( π θ ) sin 4 ( π θ ) d θ = 1 π ∫ cos 3 u sin 4 u d u = 1 π ∫ cos u ⋅ cos 2 u sin 4 u d u = 1 π ∫ cos u ⋅ ( 1 − sin 2 u ) sin 4 u d u = 1 π ∫ sin 4 u ( 1 − sin 2 u ) cos u d u Now to simplify further, we use again the u-substitution

ACHIEVE FOR CALCULUS 1 TERM >CSI CUSTOM<

4th Edition

ISBN: 9781319411657

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 7.2, Problem 25E

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To evaluate the integral ∫cos3(πθ)sin4(πθ) dθ

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Chapter 7.2, Problem 24E

Chapter 7.2, Problem 26E

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Chapter 7 Solutions

ACHIEVE FOR CALCULUS 1 TERM >CSI CUSTOM<

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To evaluate the integral ∫ cos 3 ( π θ ) sin 4 ( π θ ) d θ

Solution Summary: The author explains that to evaluate the indefinite integral, we use the u-substitution.

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