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51.

`intcos^(-1)((1-tan^(2)x)/(1+tan^(2)x)) dx = ?`A. `x^(2) + C`B. `-x^(2) + C`C. `1/(sqrt(1+x^(2))) + C`D. `1/(sqrt(1-x^(2))) + C`

Answer» Correct Answer - A
`cos^(-1)((1-tan^(2)x)/(1+ tanx^(2)x)) = cos^(-1) ((cos^(2)x - sin^(2)x)/(cos^(2)x + sin^(2)x))`
`= cos^(-1) (cos^(2)x - sin^(2)x) = cos^(-1) (cosx) = 2x`.
Discussion
52.

`int tan^(-1)(cosecx - cotx)dx = ?`A. `(x^(2))/(4) + C`B. `(-x^(2))/(4) + C`C. `(x^(2))/(2) + C`D. `(-x^(2))/(2) + C`

Answer» Correct Answer - A
`tan^(-1) (cosec x - cot x) = tan^(-1) (1/(sinx) - (cosx)/(sinx)) = tan^(-1) ((1-cosx)/(sinx))`
`= tan^(-1) {(2sin^(2)(x/2))/(2sin(x/2)cos(x/2))} = tan^(-1) tan(x/2) = x/2`.
Discussion
53.

(i) `int (cosx)/(1+cosx)dx` , (ii) `int (sinx)/((1-sinx))dx`

Answer» Correct Answer - (i) `-cosecx + cot x + x + C` , (ii) `sec x + tanx - x + C`
Discussion
54.

(i) `int(tanx)/((sec x +tanx))dx` , (ii) `int(1)/((1-sinx))dx`

Answer» Correct Answer - (i) `cot x - cosec x + C` , (ii) `tan x + sec x + C`
Discussion
55.

(i)`int (tanx + cot x)^(2) dx` , (ii) `int((1+2 sinx)/(cos^(2)x))dx` , (iii) `int ((3cosx+4)/(sin^(2)x))dx`

Answer» Correct Answer - (i) `tanx + secx + C` , (ii) `-cot x + cosec x + C`
(iii) `- 3cosec x - 4cot x + C`
Discussion
56.

`int((cotx)/(sinx) - tan^(2)x -(tanx)/(cosx) + 2/(cosx)) dx`

Answer» Correct Answer - `-cosec x+tanx+x - sec x + C`
∫(cotx/sinx - tan^2x - tanx/cosx + 2/cosx)dx
=∫(cotx.cosecx - (sec^2x - 1) - tanxsecx + 2secx)dx
= -cosecx - tanx + x - secx + 2log|secx + tanx|
Discussion