Integral Basics (continued)

Combination of substitution and trig formulas

I = ∫ sin²(x/8) dx Substitution: w = x/8, dw = dx/8, dx = 8 dw

I = 8 ∫( sin² w dw then use trig formula sin² w = ( 1 – sin 2w)/2

I = 4 ∫( 1 - sin 2w) dw which are two standard integrals

Grouping of terms (along with a simple substitution w = sec x )

I = ∫ sec² x tan x dx = ∫ sec x (sec x tan x ) dx (shows grouping)

Then use substitution to give I = ∫ w dw which is a standard integral

Substitution followed by grouping followed by another substitution

First substitution w = x/5, dw = dx/5, dx = 5 dw

I = ∫ sec² (x/5) tan (x/5) dx = 5 ∫ sec w (sec w tan w ) dw (shows grouping)

Second substitution v = sec w , dv = sec w tan w dw

Then use substitution to give I = ∫ v dv with again is a standard integral

Integration by parts is very useful for integrals of the form I = ∫ u dv.

Then the integral, I , becomes

I = u v - ∫ v du i.e. Apply to the integral I = ∫ x cos x dx

Here u = x and dv = cos x dx

Then u = x and dv = cos x dx

du = dx and v = sin x

So I = x sin x - ∫ sin x dx = x sinx + cos x + C (result)

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