Integration

Worked examples (6)

• 1Integrating sinh and cosh directly
• 2Recognising a standard inverse-form integral
• 3Completing the square to reach a tan⁻¹ form
• 4A coefficient on x² under a square root
• 5A definite integral giving cosh⁻¹
• 6Negative x² coefficient — complete the square then sin⁻¹

Model answers (6)

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Worked examples (6)

• 1Using a given reduction formula
• 2Deriving the reduction formula for ∫xⁿeˣ
• 3Recursive evaluation of a definite integral
• 4Two-index integral — reducing one parameter
• 5Deriving the sinⁿx reduction formula
• 6Wallis-type evaluation of ∫sin⁶x

Model answers (6)

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Worked examples (6)

• 1Estimating area with rectangles (Riemann sum)
• 2Area under y = 1/x as a logarithm
• 3Region partly below the x-axis
• 4Area using an inverse-hyperbolic standard form
• 5Exact area giving a multiple of π
• 6Area measured to the y-axis

Model answers (6)

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Worked examples (6)

• 1Arc length of a simple curve
• 2Arc length of a catenary y = cosh x
• 3Parametric arc length
• 4Surface of revolution about the x-axis
• 5Surface area from a catenary
• 6Surface of revolution about the y-axis

Model answers (6)

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• ✓Model answer 6