Question

5.sin (ax +b) cos (ax +b)

Answer 1

The integral of the function is given as,

sin( ax+b )cos( ax+b ) (1)

Multiply and divide equation (1) by 2

sin( ax+b )cos( ax+b )= 2sin( ax+b )cos( ax+b ) 2 = sin2( ax+b ) 2 (2)

Consider 2( ax+b )=t .

2ax+2b=t

Differentiating both sides, 2adx=dt

Substitute 2adx=dt in equation (2) and then integrate,

∫ sin2( ax+b ) 2 dx= 1 2 ∫ sint 2a dt = 1 4a ∫ sintdt = 1 4a [ −cost ]+C = −1 4a cos2( ax+b )+C

Thus, the required integral is I= −1 4a cos2( ax+b )+C .