(i) tan –1 + C (ii) tan –1 + C

(iii) x –  n |3cosx + 2sinx| + C (iv)  n + C

(v) tan –1 + C (vi) + C

(vii) tan x + sin 2x − + C

Sol. (i) Ι =

= = put tan = t

⇒ sec 2 dx = dt ⇒ Ι = = tan –1 + C

= tan -1 + C

(ii) Ι =

=

= put tan = t ⇒ sec 2 dx = dt

⇒ Ι =

= tan -1 (t) + C = tan –1 + C

(iii) Ι =

put 2 sin x + 2 cos x = A(3 cos x + 2 sin x) + B(–3 sin x + 2 cos x)

⇒ 2 = 3A + 2B and 2 = 2A – 3B

⇒ A = B =

⇒ Ι = = –

= (x) – ln |(3 cos x + 2 sin x)| + C

(iv) Ι =

= put tan x/2 = t ⇒ sec 2 (x/2) dx = dt

Ι = =  n |t + 1| + C =  n + C

(v) Ι = = put tanx = t sec 2 x dx = dt

Ι = = = + C =

(vi) Ι = = put 1 – cot x = t ⇒ cosec 2 x dx = dt

⇒ Ι =  n |t| + C =  n |(1 – cot x)| + C

(vii) = =

= =

=

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