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Calculus Level 3

If ∫ d x 2 a x − x 2 = a n sin ⁡ − 1 [ x a − 1 ] \displaystyle \large\int { \cfrac { dx }{ \sqrt { 2ax-{ x }^{ 2 } } } } ={ a }^{ n }\sin ^{ -1 }{ \left[ \cfrac { x }{ a } -1 \right] }

Find the value of n \displaystyle \large n

AYWC?
none of these -1 0 1

Soumo Mukherjee by Soumo Mukherjee , 25, India

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1 solution

Soumo Mukherjee
Apr 7, 2015

∫ d x 2 a x − x 2 \displaystyle \int { \cfrac { dx }{ \sqrt { 2ax-{ x }^{ 2 } } } } is dimensionless. Therefore a n sin ⁡ − 1 [ x a − 1 ] \displaystyle { a }^{ n }\sin ^{ -1 }{ \left[ \cfrac { x }{ a } -1 \right] } must also be dimensionless (Both RHS & LHS must have identical dimensions). And that is possible only if n = 0 n=0 (i.e, a 0 sin ⁡ − 1 [ x a − 1 ] = sin ⁡ − 1 [ x a − 1 ] { a }^{ 0 }\sin ^{ -1 }{ \left[ \cfrac { x }{ a } -1 \right] } =\sin ^{ -1 }{ \left[ \cfrac { x }{ a } -1 \right] } . Recall sin ⁡ − 1 y \sin ^{ -1 }{ y } represents a number. Numbers have no dimensions.)

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Please refrain from posting anti solutions as official solutions.

Abhishek Sharma - 6 years, 1 month ago

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what do you mean?

Soumo Mukherjee - 6 years, 1 month ago

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