If x ≠ (2n+1) \,π/2, then ³ x/(1+ x)^4 \, dx =

Correct Answer is : None of the above

Correct Answer is : None of the above

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TS EAMCET 20-July-2022 Shift 1 Solved Paper

Section: Mathematics

Question : 75 of 160

Marks: +1, -0

x \neq (2n+1) \,\frac{\pi}{2}

\int \frac{\cos^3 x}{(1+\sin x)^4} \, dx =

Solution: 👈: Video Solution

Given,

x \neq (2n+1) \,\frac{\pi}{2}

Now,

\int \frac{\cos^3 x}{(1+\sin x)^4} \, dx = \int \frac{\cos x (1-\sin^2 x)}{(1+\sin x)^4} \, dx

On putting

1+\sin x = t

\cos x \, dx = dt

\int \frac{1-(t-1)^2}{t^4} \, dt = \int \frac{-t^2+2t}{t^4} \, dt

= \int \left(\frac{2}{t^3} - \frac{1}{t^2}\right) dt = -\frac{1}{t^2}+\frac{1}{t}+c

= \frac{t-1}{t^2}+c\, dt = \frac{\sin x}{(1+\sin x)^2}+c

The given options are wrong in this question. The correct answer is

\frac{\sin x}{(1+\sin x)^2}+C

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