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    i dont understand this method to work out ∫x/x^2+1

    solution: 1/2 ∫ 2x/ x^2+1 dx = 1/2 In|x^2 + 1| + c.

    Do I need to know this method for AQA C3 or should I just use integration by substitution.

    In addition, can someone help me prove ∫cot x = In|sin x| + c
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    (Original post by man111111)
    i dont understand this method to work out ∫x/x^2+1

    solution: 1/2 ∫ 2x/ x^2+1 dx = 1/2 In|x^2 + 1| + c.

    Do I need to know this method for AQA C3 or should I just use integration by substitution.

    In addition, can someone help me prove ∫cot x = In|sin x| + c
    Yes substitution is fine but you need to know what to substitute because you might not be given it for a question like this.


    Do do it their way, consider the derivative of \ln(x^2+1). This is

    \dfrac{2x}{x^2+1}

    This means that

    \displaystyle \int \frac{2x}{x^2+1} \ dx = ln (x^2+1) + c

    So then if you adjust the constant you end up with

    \displaystyle \int \frac{x}{x^2+1} \ dx = \frac{1}{2}\ln (x^2+1) + c
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    (Original post by man111111)
    i dont understand this method to work out ∫x/x^2+1

    solution: 1/2 ∫ 2x/ x^2+1 dx = 1/2 In|x^2 + 1| + c.

    Do I need to know this method for AQA C3 or should I just use integration by substitution.

    In addition, can someone help me prove ∫cot x = In|sin x| + c
    It is a handy trick to know, but not a ‘must’ for C3.
    All they did was mult the numerator by 2 and place a half outside (this does not change the integral since 2 and 1/2 would cancel) and the 2x on the numerator is precisely the derivative of the denominator hence you can just interate straight to natural log as shown.


    For cot, note that cot=cos/sin where the numerator is the derivative of the denominator. Hence natural log
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    (Original post by RDKGames)
    It is a handy trick to know, but not a ‘must’ for C3.
    All they did was mult the numerator by 2 and place a half outside (this does not change the integral since 2 and 1/2 would cancel) and the 2x on the numerator is precisely the derivative of the denominator hence you can just interate straight to natural log as shown.


    For cot, note that cot=cos/sin where the numerator is the derivative of the denominator. Hence natural log
    ok thank you
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    (Original post by Notnek)
    Yes substitution is fine but you need to know what to substitute because you might not be given it for a question like this.


    Do do it their way, consider the derivative of \ln(x^2+1). This is

    \dfrac{2x}{x^2+1}

    This means that

    \displaystyle \int \frac{2x}{x^2+1} \ dx = ln (x^2+1) + c

    So then if you adjust the constant you end up with

    \displaystyle \int \frac{x}{x^2+1} \ dx = \frac{1}{2}\ln (x^2+1) + c
    thanks
 
 
 

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