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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
Question
Integrate the functions in Exercises:
$\frac{\cos\text{x}}{\sqrt{1+\sin\text{x}}}$
✓

Answer

$\text{Let I}=\int\frac{\cos\text{x}}{\sqrt{1+\sin\text{x}}}\text{dx} $
Putting $1 +\sin\text{x}=\text{t}\ \ \ \ \Rightarrow\ \ \ \ \cos\text{x}=\frac{\text{dt}}{\text{dx}}\ \ \ \ \Rightarrow\ \ \ \cos\text{ x dx = dt} $
$\therefore \ \ \ \ \ $From eq. (i), $\text{I}=\int\frac{\text{dt}}{\sqrt{\text{t}}}=\int\text{t}^{\frac{-1}{2}}\text{ dt}=\frac{\text{t}^{\frac{-1}{2}+1}}{\frac{-1}{2}+1}+\text{c}$
$=\frac{\text{t}^{^1/_2}}{^1/_2}+\text{c} =2\sqrt{t} +\text{c}=2\sqrt{1+\sin\text{x}}+\text{c}$

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