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Byju's Answer

Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

∫√1+cos xdx=

Question

$\int \sqrt{(1 + \cos x)} d x =$

$2 \sin (\frac{x}{2}) + c$

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$\sqrt{2} \sin (\frac{x}{2}) + c$

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$(\frac{1}{2}) \sin (\frac{x}{2}) + c$

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$(\frac{\sqrt{2}}{2}) \sin (\frac{x}{2}) + c$

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$2 \sqrt{2} \sin (\frac{x}{2}) + c$

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Solution

The correct option is E
$2 \sqrt{2} \sin (\frac{x}{2}) + c$

Explanation for correct option:
Find the value of given integration:
Consider the given Equation as
$I = \int \sqrt{(1 + \cos x)} d x$
We know that,
$(1 + \cos x) = 2 \cos^{2} (\frac{x}{2})$
Then,
$I = \int \sqrt{(1 + \cos x)} d x = \int \sqrt{2 \cos^{2} (\frac{x}{2})} d x = \sqrt{2} \int \cos (\frac{x}{2}) d x = \sqrt{2} \frac{\sin (\frac{x}{2})}{(\frac{1}{2})} + c I = 2 \sqrt{2} \sin (\frac{x}{2}) + c$
Hence, the correct answer is option (E).

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∫1+cosxdx=

The correct option is E 22sinx2+cExplanation for correct option:Find the value of given integration:Consider the given Equation asI=∫1+cosxdxWe know that,1+cosx=2cos2x2Then,I=∫1+cosxdx=∫2cos2x2dx=2∫cosx2dx=2sinx212+cI=22sinx2+cHence, the correct answer is option (E).

$\int \sqrt{(1 + \cos x)} d x =$