1.sin2 (2x +5)

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Solution

The given function is sin 2 ( 2x+5 )

∫ sin 2 ( 2x+5 )dx (1)

Also, sin 2 ( y )= 1−cos( 2y ) 2 (2)

Compare (1) and (2), we get,

y=2x+5

Substitute value of y in (2).

sin 2 ( 2x+5 )= 1−cos[ 2( 2x+5 ) ] 2 = 1−cos( 4x+10 ) 2

Substitute value of sin 2 ( 2x+5 ) in (1).

∫ sin 2 ( 2x+5 )dx = ∫ 1−cos( 4x+10 ) 2 = 1 2 ∫ 1dx − 1 2 ∫ cos( 4x+10 )dx = x 2 − 1 2×4 sin( 4x+10 )+c = x 2 − 1 8 sin( 4x+10 )+c

Thus, the integral of the function sin 2 ( 2x+5 ) is x 2 − 1 8 sin( 4x+10 )+c.

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