Advanced Integration Techniques and Examples
Explore advanced integration methods, such as integration by inspection, standard function integration, and finding definite integrals. Dive into differentiation results, chain rules, and calculating various integrals step by step. Enhance your understanding of integration theory and practice with detailed explanations and examples.
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Presentation Transcript
Integration By inspection
Integration by Inspection BAT Integrate using standard results from differentiation BAT Integrate using the reverse of the chain rule KUS objectives Starter: True or False Integration is the reverse process of Differentiation Integration does not give a unique answer A definite Integral finds the area under a graph between the function, and bounds a and b on the x-axis
WB1integrate standard functions You met the following in differentiation: ?? ??= ? ?? 1 a) ? = ?? ?? ??= e) ? = ?? ?? ?? ??= 4(1 + ?)3 b) ? = (1 + ?)4 ?? ??= 1 ? f) ? = ln? c) ? = (3? + 2)4 g) ? =?3?+2 ?? ??= ?? ??= 12(3? + 2)3 3?3?+2 1 12(3? + 2)4 h) ? = ln(3? + 2) d) ? = 3 ?? ??= ?? ??= (3? + 2)3 3? + 2
Therefore, you already can deduce the following WB 1b 1 12(3? + 2)4 d) ? = 1 12(3? + 2)4+ C (3? + 2)3?? = ?? ??= (3? + 2)3 3 ) ? = ln(3? + 2) 3? + 2?? = ln (3x + 2) + C ?? ??= 3 3? + 2
WB 2aFind the following Integrals a) 2? + 44?? b) 18 4? 58?? c) 21 3? + 36?? a) Think of ? ??2? + 45 by chain rule = 10 2? + 44 As this is 10 times what we want, we need to divide our guess by 10 1 2? + 44?? = 102? + 45+ ?
WB 2bcFind the following Integrals a) 2? + 44?? b) 18 4? 58?? c) 21 3? + 36?? b) Think of ? ??4? 59 by chain rule = 36 4? 58 1 24? 59+ ? 18 4? 58?? = c) Think of ? ??3? + 37 by chain rule = 21 3? + 36 3? + 37+? 21 3? + 36?? =
WB3abyou can also deduce the following g) ? =?3?+2 ?) ?3?+2?? =1 3?3?+2+ C ?? ??=3?3?+2 =1 4?4?+1+ C b) ?4?+1 ??
WB3cd Find the following Integrals c) 1 d) 2 3?6? 5 ?? 3?9?+2 ?? 3?6? 5 ?? =1 ?) 1 3?3?+2+ C 3?9?+2 ??=1 d) 2 4?4?+1+ C
These answers illustrate a rule: Notes ?4?+1?? 2? + 44?? =1 1 4?4?+1 102? + 45 = 1) Integrate the function using what you know from differentiation 2) Divide by the coefficient of x 3) Simplify if possible and add C This technique will only work for linear transformations of functions such as f(ax + b)
KUS objectives BAT Integrate using standard results from differentiation BAT Integrate using the reverse of the chain rule self-assess One thing learned is One thing to improve is
