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Evaluate the integral of 1/ ( 1 + 4x^2)  

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wondergirl | Student, Grade 10 | eNoter

Posted January 27, 2011 at 11:38 AM via web

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Evaluate the integral of 1/ ( 1 + 4x^2)

 

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted January 27, 2011 at 11:45 AM (Answer #1)

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We have to find Int [1/ (1 + 4x^2) dx].

First substitute u = 2x

=> du /dx = 2

=> du /2 = dx

Now Int [1/ (1 + 4x^2) dx]

=> Int [(1/2)*(1/ (1+u^2) du]

=> (1/2)*Int [1/ (1 + u^2) du]

Now Int [1/ (1+u^2) du] =arc tan u + C

=> (1/2)*arc tan u + C

replace u with 2x

=> (1/2)* arc tan 2x + C

Therefore the required result is

1/2)* arc tan 2x + C

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neela | High School Teacher | Valedictorian

Posted January 27, 2011 at 12:10 PM (Answer #2)

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To evaluate the integral of 1/(1+4x^2).

Solution:

We know that Integral of 1/(x^2+a^2 ) =  Int {1/(x^2+a^2)} dx = (1/a)arc tan (x/a) + C....(1) .

The given function 1/(4x^2+1) = (1/4){1/[x^2+(1/4)]} =  (1/4){1/[x^2+(1/2)^2]}.

Therefore Int {1/(4x^2+1)} dx = (1/4) Int {1/[x^2+(1/2)^2]} dx which is like (1). Here a = 1/2.

So Int {1/(4x^2+1)} dx = (1/4)* (1/(1/2))* arc tan {x/(1/2)}+C .

Therefore Int {1/(4x^2+1)} dx = (1/2) arc tan(2x) +C.

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