Question

tanx/3+2tan^2(x) find integration

Answer 1

I = ∫ [ tan x / ( 3 + 2 tan² x ) ] dx

= ∫ { ( sin x / cos x ) / [ 3 + 2( sin² x / cos² x ) ] } dx

= ∫ [ ( sin x cos x ) / ( 3 cos² x + 2 sin² x ) ] dx

= ∫ [ 1 / ( 2 sin² x + 3 cos² x ) ] • sin x cos x dx ................. (1)

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Let : u = 2 sin² x + 3 cos² x.

Then :

du/dx = 2. 2( sin x )( cos x ) + 3. 2( cos x )( - sin x )

. . . . .= 4 sin x cos x - 6 sin x cos x

. . . . .= -2 sin x cos x

Hence : sin x cos x dx = (-1/2) du.