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tatyana61 [14]
3 years ago
7

Evaluate ∫ sin(x) cos(x) dx by four methods.

Mathematics
1 answer:
Sliva [168]3 years ago
3 0

(a) Since <em>u</em> = cos(<em>x</em>) gives d<em>u</em> = -sin(<em>x</em>) d<em>x</em>, we have

∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = - ∫ (-sin(<em>x</em>)) cos(<em>x</em>) d<em>x</em>

= - ∫ cos(<em>x</em>) d(cos(<em>x</em>))

= - ∫ <em>u</em> d<em>u</em>

= - 1/2 <em>u</em>² + C

= -1/2 cos²(<em>x</em>) + C

(b) Now <em>u</em> = sin(<em>x</em>) gives d<em>u</em> = cos(x) d<em>x</em>, so

∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = ∫ sin(<em>x</em>) d(sin(<em>x</em>))

= ∫ <em>u</em> d<em>u</em>

= 1/2 <em>u</em>² + C

= 1/2 sin²(<em>x</em>) + C

(c) Since sin(2<em>x</em>) = 2 sin(<em>x</em>) cos(<em>x</em>), we have

∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = 1/2 ∫ sin(2<em>x</em>) d<em>x</em>

Substitute <em>u</em> = 2<em>x</em>, so that d<em>u</em> = 2 d<em>x</em>, and

∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = 1/2 ∫ sin(2<em>x</em>) d<em>x</em>

= 1/4 ∫ 2 sin(2<em>x</em>) d<em>x</em>

= 1/4 ∫ sin(<em>u</em>) d<em>u</em>

= -1/4 cos(<em>u</em>) + C

= -1/4 cos(2<em>x</em>) + C

(d) Integrate by parts, setting

<em>u</em> = sin(<em>x</em>)          ==>  d<em>u</em> = cos(<em>x</em>) d<em>x</em>

d<em>v</em> = cos(<em>x</em>) d<em>x</em>  ==>  <em>v</em> = sin(<em>x</em>)

Then

∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = sin²(<em>x</em>) - ∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em>

2 ∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = sin²(<em>x</em>) + C

∫ sin(<em>x</em>) cos(<em>x</em>) d<em>x</em> = 1/2 sin²(<em>x</em>) + C

The solutions in (b) and (d) are identical, but all 4 are equivalent, and this follows from the identities,

sin²(<em>x</em>) + cos²(<em>x</em>) = 1

cos(2<em>x</em>) = cos²(<em>x</em>) - sin²(<em>x</em>)

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