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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS2 Marks
Question
By Using properties of definite integral, evaluate the following integral in Exercise:
$\int^{\frac{\pi}{2}}_{0}\frac{\sin\text{x}-\cos\text{x}}{1+\sin\text{x}\cos\text{x}}\text{dx}$
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Answer

$\text{Let}\text{I}=\int^{\frac{\pi}{2}}\limits_{0}\frac{\sin\text{x}-\cos\text{x}}{1+\sin\text{x}\cos\text{x}}\text{dx}$
$\Rightarrow\ \ \text{I}=\int^{\frac{\pi}{2}}\limits_{0}\frac{\sin\bigg(\frac{\pi}{2}-\text{x}\bigg)-\cos\bigg(\frac{\pi}{2}-\text{x}\bigg)}{1+\sin\bigg(\frac{\pi}{2}-\text{x}\bigg)\cos\bigg(\frac{\pi}{2}-\text{x}\bigg)}\text{dx}=\int^{\frac{\pi}{2}}\limits_{0}\frac{\cos\text{x}-\sin\text{x}}{1+\cos\text{x}\sin\text{x}}\text{dx}=-\int^{\frac{\pi}{2}}\limits_{0}\frac{\sin\text{x}-\cos\text{x}}{1+\sin\text{x}\cos\text{x}}\text{dx}$
Adding eq. (i) and (ii), we have         $21=0 \ \ \Rightarrow 1=0$

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