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Question: \(\int_{}^{}\left( \frac{dx}{\sin x + \sec x} \right)\)is –...

∫(dxsin⁡x+sec⁡x)\int_{}^{}\left( \frac{dx}{\sin x + \sec x} \right)is –

A

123\frac{1}{2\sqrt{3}}ln∣tan⁡x+3tan⁡x−3∣\left| \frac{\tan x + \sqrt{3}}{\tan x - \sqrt{3}} \right| + tan–1 (sin x + cos x) + c

B

123\frac{1}{2\sqrt{3}}ln∣sin⁡x−cos⁡x+3sin⁡x−cos⁡x−3∣\left| \frac{\sin x - \cos x + \sqrt{3}}{\sin x - \cos x - \sqrt{3}} \right| + tan–1 (sin x + cos x) + c

C

123\frac{1}{2\sqrt{3}}ln∣tan⁡x+3tan⁡x−3∣\left| \frac{\tan x + \sqrt{3}}{\tan x - \sqrt{3}} \right| + tan–1 (sin x – cos x) + c

D

None of these

Answer

123\frac{1}{2\sqrt{3}}ln∣sin⁡x−cos⁡x+3sin⁡x−cos⁡x−3∣\left| \frac{\sin x - \cos x + \sqrt{3}}{\sin x - \cos x - \sqrt{3}} \right| + tan–1 (sin x + cos x) + c

Explanation

Solution

I = ∫dx(sin⁡x+sec⁡x)\int \frac { d x } { ( \sin x + \sec x ) } =

= 12\frac { 1 } { 2 } ∫cos⁡x+sin⁡x1+sin⁡xcos⁡x\int \frac { \cos x + \sin x } { 1 + \sin x \cos x }dx + 12\frac { 1 } { 2 } ∫cos⁡x−sin⁡x1+sin⁡xcos⁡x\int \frac { \cos x - \sin x } { 1 + \sin x \cos x }dx

Let sin x – cos x = t Let sin x + cos x = u

Ž (cos x + sin x) dx = dt Ž (cos x – sin x) dx = du

So I = 12\frac { 1 } { 2 } ∫dt1+(1−t22)\int \frac { \mathrm { dt } } { 1 + \left( \frac { 1 - \mathrm { t } ^ { 2 } } { 2 } \right) } +12\frac { 1 } { 2 }

= ∫dt3−t2\int \frac { \mathrm { dt } } { 3 - \mathrm { t } ^ { 2 } } += 123\frac { 1 } { 2 \sqrt { 3 } }ln ∣t+3t−3∣\left| \frac { t + \sqrt { 3 } } { t - \sqrt { 3 } } \right| + tan–1 u + c

= 123\frac { 1 } { 2 \sqrt { 3 } }ln ∣sin⁡x−cos⁡x+3sin⁡x−cos⁡x−3∣\left| \frac { \sin x - \cos x + \sqrt { 3 } } { \sin x - \cos x - \sqrt { 3 } } \right|+tan–1 (sin x + cos x) + c

\ (2) is correct