_{π/11}^{9π/22} \;{dx}{1+{ x}} =

Correct Answer is : 7π447π/44447π

Correct Answer is : 7π447π/44447π

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AP EAPCET 08-Jul-2022 Shift 1 Paper

Section: Mathematics

Question : 74 of 160

Marks: +1, -0

\int\limits_{\frac{\pi}{11}}^{\frac{9\pi}{22}} \;\frac{dx}{1+\sqrt{\tan x}} =

Solution: 👈: Video Solution

Let

I = \int\limits_{\frac{\pi}{11}}^{\frac{9\pi}{22}} \frac{1}{1+\sqrt{\tan x}}\,dx

= \int\limits_{\frac{\pi}{11}}^{\frac{9\pi}{22}} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx

\dots(i)

= \int_{\frac{\pi}{11}}^{\frac{9\pi}{22}} \frac{\sqrt{\cos x\left(\frac{9\pi}{22}+\frac{\pi}{11}-x\right)}}{\sqrt{\sin\left(\frac{9\pi}{22}+\frac{\pi}{11}-x\right) + \sqrt{\cos\left(\frac{9\pi}{22}+\frac{\pi}{11}-x\right)}}}\,dx

= \int_{\frac{\pi}{11}}^{\frac{9\pi}{22}} \frac{\sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}}\,dx

\dots(ii)

On adding both Eqs. (i) and (ii),

2I = \int_{\frac{\pi}{11}}^{\frac{9\pi}{22}} dx = \left[ x \right]_{\frac{\pi}{11}}^{\frac{9\pi}{22}}

= \frac{9\pi}{22} - \frac{\pi}{11}

I = \frac{1}{2} \left[ \frac{7\pi}{22} \right] = \frac{7\pi}{44}

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