If -0-dx-a-cos x-a2-1- then the value of -0-dx-10-cos x3 is equal to where -a-1

∫_0^πdx/(a cos x)=π/√(a^2 1) Differentiating both sides with respect to a, we get ∫_0^πdx/(a cos x)^2= π a/(a^2 1)^3/2 Again differentiating with respect to a, ...

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Standard XII

Mathematics

Euler's Representation

If ∫0πdx/a-co...

Question

If $π \int 0 \frac{d x}{a - cos x} = \frac{π}{\sqrt{a^{2} - 1}}$ , then the value of $π \int 0 \frac{d x}{(\sqrt{10} - cos x)^{3}}$ is equal to (where $| a | > 1$ )

\frac{π}{81}

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\frac{7 π}{162}

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\frac{7 π}{81}

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\frac{π}{162}

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Solution

The correct option is B $\frac{7 π}{162}$
$π \int 0 \frac{d x}{(a - cos x)} = \frac{π}{\sqrt{a^{2} - 1}}$
Differentiating both sides with respect to $a$ , we get
$- π \int 0 \frac{d x}{(a - cos x)^{2}} = \frac{- π a}{(a^{2} - 1)^{\frac{3}{2}}}$
Again differentiating with respect to $a,$ we get
$2 π \int 0 \frac{d x}{(a - cos x)^{3}} = \frac{π (1 + 2 a^{2})}{(a^{2} - 1)^{\frac{5}{2}}}$
Putting $a = \sqrt{10},$ we get
$π \int 0 \frac{d x}{(\sqrt{10} - cos x)^{3}} = \frac{7 π}{162}$

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Euler's Representation

Standard XII Mathematics

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If π∫0dxa−cosx=π√a2−1, then the value of π∫0dx(√10−cosx)3 is equal to (where |a|>1)

The correct option is B 7π162π∫0dx(a−cosx)=π√a2−1 Differentiating both sides with respect to a, we get −π∫0dx(a−cosx)2=−πa(a2−1)32 Again differentiating with respect to a, we get 2π∫0dx(a−cosx)3=π(1+2a2)(a2−1)52 Putting a=√10, we get π∫0dx(√10−cosx)3=7π162

If $π \int 0 \frac{d x}{a - cos x} = \frac{π}{\sqrt{a^{2} - 1}}$ , then the value of $π \int 0 \frac{d x}{(\sqrt{10} - cos x)^{3}}$ is equal to (where $| a | > 1$ )