Evaluate Integral from nothing to nothing StartFraction 5 dx Over x squared minus 4 EndFraction , for x greater than 2, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

Step-by-step explanation:

Partial fraction:

Given integration is

(x²-4)= (x+2)(x-2)

Let

Therefore,

A+B=0 ⇒2A+2B=0...(1)        and       2B-2A=1.....(2)

Adding (1) and (2)

2A+2B+2B-2A=0+1

⇒4B=1

Putting the value of B in equation (1) we get

Therefore,

     [ c is arbitrary constant]

Trigonometric substitution:

Given integration is

Let x = 2 sinθ

⇒dx = 2 cosθ dθ

   [ multiply the numerator and denominator by ]

let ,

  [ c is arbitrary constant]

Since

Putting the value of secθ and tanθ

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step-by-step explanation:

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step-by-step explanation: