Page 188 - ADVANCED MATH F.5

Form Five Form Five

Integration by parts
Teaching steps

1. Guide students to discuss the type of integrands (which
consists two functions) for integrals that can be determined
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by integrating it by parts. Assist them through discussion to
identify products of two functions, where the left part of the
integrand is considered as the first function and its right part as
the second function.
2. Lead students through discussion to derive the method
of integration by parts given ()ux and ()vx are any two
differentiable functions. Assist students in obtaining the
integration by parts method which is given by,
⌠  u ()x d vx u () ()x vx −  ⌠ vx d u ()x dx or udv uv= − ∫ vdu
∫
()
()dx =
⌡ dx ⌡ dx
3. Guide students to choose the function whose derivative is
easily integrated. Assist them to use the order of preference
ILATE, where I stands for inverse trigonometric functions,
L stands for the logarithmic functions, A stands for algebraic
functions, T stands for the trigonometric functions, and E
stands for exponential functions.

4. Engage students through group discussions to apply integration
by parts for integrals of the form:

∫
( )dx
( )dx
(a) x n sin ax or ∫ x n cos ax .
∫
(b) x n ln ( ),ax where a is a constant
∫
∫
(c) x n sin( )ax dx or x n cos( ) ,ax dx where a is a constant
∫
(d) e ax sin bx ∫ ax cos bx
( )dx or e
( ) dx, where a and b are
constants
∫
∫
∫
1
−
1
1
−
(e) x n sin x dx, x n cos x dxor x n tan x dx
−
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