1.

Evaluate dy/dx = tan2 (x + y).

Answer»

dy/dx = tan2 (x + y)

put v = x + y

dv/dx = 1 + (dy/dx) = 1 + tan2 v = sec2 v

∫ dv/sec2 v = ∫ dx

= ∫ cos2 v.dv = x + c

∫ (1 + cos 2v/2) dv = x + c

∫ (1 + cos 2v) dv = 2x + 2c 

v + (sin 2v/2) = 2x + 2c 

2v + sin 2v = 4x + c' 

2(x + y) + sin 2(x + y) = 4x + c' 

x – y – (1/2) sin [2(x + y)] = c


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