-2 -0 x In 3-cos x-3-cos xdx -

I = ∫^2 π_0 (2 π x) In (3+cos x/3 cos x)dx ⇒ 2I = 2 π∫_0^2 π In (3 + cos x/3 cos x)dx ⇒ I= 4 π∫^π_0 In (3 + cosx/3 cosx)dx = 4 π∫^π_0 In (3 + cos x/3 cos ...

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Byju's Answer

Standard XIII

Mathematics

Inequalities of Integrals

∫2 π0 x In 3+...

Question

$\int_{0}^{2 π} x I n (\frac{3 + c o s x}{3 - c o s x}) d x =$ ___

\frac{π}{2} I n 3

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\frac{π}{6} I n 3

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\frac{π}{12} I n 3

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Solution

The correct option is D $0$
$I = \int_{0}^{2 π} (2 π - x) I n (\frac{3 + c o s x}{3 - c o s x}) d x$
$\Rightarrow 2 I = 2 π \int_{0}^{2 π} I n (\frac{3 + c o s x}{3 - c o s x}) d x$
$\Rightarrow I = 4 π \int_{0}^{π} I n (\frac{3 + c o s x}{3 - c o s x}) d x = 4 π \int_{0}^{π} I n (\frac{3 + c o s x}{3 - c o s x}) d x = - I$
$\Rightarrow I = 0$

Suggest Corrections

∫2π0x In(3+cosx3−cosx)dx=___

The correct option is D 0I=∫2π0(2π−x) In (3+cosx3−cosx)dx ⇒2I=2π∫2π0In(3+cosx3−cosx)dx ⇒I= 4π∫π0In(3+cosx3−cosx)dx=4π∫π0In(3+cosx3−cosx)dx=−I ⇒I=0

$\int_{0}^{2 π} x I n (\frac{3 + c o s x}{3 - c o s x}) d x =$ ___