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    yooo so my dumb mind thought the integral of 1/x^2 was ln(x^2) but it's not, and can anyone explain why its -1/x instead?? Lol I know it's a stupid question but pls help. THanks
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    (Original post by btsseokjin)
    yooo so my dumb mind thought the integral of 1/x^2 was ln(x^2) but it's not, and can anyone explain why its -1/x instead?? Lol I know it's a stupid question but pls help. THanks
    \displaystyle \frac{1}{x^2} = x^{-2}

    so this is an integral you can do with C1 methods.

    If you need help understanding why it's not ln(x^2), please explain to us why you think it is ln(x^2).
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    My bad....
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    The integral of f'(x)/f(x) would be ln|f(x)|. You probably thought this was an integral of 1/x which is ln|x| or you tried to follow the technique but it won't work. You can use U-sub to show integral of f'(x)/f(x) would be ln|f(x)|. In this case, integral of 1/x fits the bill e.g. 1=d/dx[x] so you get lnx. It won't work here. This uses indices and the C1 integral rule you learnt.
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    1/x^2 = x^-2
    integrate to get (1/-1)*x^-1 = - 1/x

    You can't integrate it to ln(x^2) as differentiating ln(x^2) gives you 2x/x^2 = 2/x (chain rule)
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    Law of indices. Convert 1/x^2 to x^-2 and integrate that. To integrate with respect to x at 1 to the exponent and divide by the new exponent. You'll get the answer.
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    (Original post by Notnek)
    \displaystyle \frac{1}{x^2} = x^{-2}

    so this is an integral you can do with C1 methods.

    If you need help understanding why it's not ln(x^2), please explain to us why you think it is ln(x^2).
    omg why didn't i think of that😂 that's a lot simpler than what I imagined😂

    I thought it was ln(x^2) cuz the integral of 1/x is ln(x) and I just thought it was the same lol

    Thank you tho!!!!!
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    (Original post by Grizzelex)
    Law of indices. Convert 1/x^2 to x^-2 and integrate that. To integrate with respect to x at 1 to the exponent and divide by the new exponent. You'll get the answer.

    Thank you!!! Much appreciated!
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    (Original post by btsseokjin)
    Thank you!!! Much appreciated!
    No problem, are you prepping for A levels? If so, good luck with your exams. Just don't give up.
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    (Original post by btsseokjin)
    yooo so my dumb mind thought the integral of 1/x^2 was ln(x^2) but it's not, and can anyone explain why its -1/x instead?? Lol I know it's a stupid question but pls help. THanks
    Integration is concerned with finding the area under a curve while differentiation is concerned with the gradient of a curve. More fundamentally, we can link the two as inverse operations using the Fundamental Theorem of Calculus. In fact, another name for an integral is an antiderivative.

    What this means in particular is that if the integral of a function, f, is F, then the derivative of F is F' = f. Linking this to your example, if you differentiate ln(x²) and -1/x, you should be able to argue which is truly an integral of 1/x²dx.
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    (Original post by btsseokjin)
    omg why didn't i think of that😂 that's a lot simpler than what I imagined😂

    I thought it was ln(x^2) cuz the integral of 1/x is ln(x) and I just thought it was the same lol

    Thank you tho!!!!!
    It's very common for C3/4 students to not notice the simpler integrals! I advise always checking first if an integral can be done simply using C1 methods before looking for a harder method.

    Also, "I just thought it was the same" is not a good thought process for integration You need to known when/why all of the integration methods can be used.
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    (Original post by Grizzelex)
    No problem, are you prepping for A levels? If so, good luck with your exams. Just don't give up.
    I am indeed!! Thank you!!!
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    Yo I thought the same then I realized it's a C1 integral lmao
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    (Original post by MR1999)
    Integration is concerned with finding the area under a curve while differentiation is concerned with the gradient of a curve. More fundamentally, we can link the two as inverse operations using the Fundamental Theorem of Calculus. In fact, another name for an integral is an antiderivative.

    What this means in particular is that if the integral of a function, f, is F, then the derivative of F is F' = f. Linking this to your example, if you differentiate ln(x²) and -1/x, you should be able to argue which is truly an integral of 1/x²dx.
    Yeah i can see that ln(x^2) is the wrong integral as the differential is 2/x. Thank you!!
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    (Original post by Notnek)
    It's very common for C3/4 students to not notice the simpler integrals! I advise always checking first if an integral can be done simply using C1 methods before looking for a harder method.

    Also, "I just thought it was the same" is not a good thought process for integration You need to known when/why all of the integration methods can be used.

    Yeah that would be a smart idea😂.
    I agree, I gotta revise more in that area tbf😂
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    (Original post by Sanjith Hegde123)
    Yo I thought the same then I realized it's a C1 integral lmao
    Dude ikr, mans gotta do more revision 😂
 
 
 
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