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Math Help - A few integration problems

  1. #1
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    A few integration problems

    I am not supposed to "integrate by substitution" or anything fancy yet in this section ......

    "In each part, confirm that the formula is correct and state a corresponding integration formula."

    a) (d/dx)[sqrt(1+ X2)] = (X)(1+X2)-1/2

    Integral
    (X)(1+X2)-1/2 dx

    I understand how to take the integral of (1+X2)-1/2 , but where does the x go? is it x(dx) and becomes 1? and then whats leftover is the integral of (X)(1+X2)-1/2 ??? That would make sense, I just figured that out while I was typing it but I wanted to make sure ...

    b) (d/dx)[1/3 sin(1+x3)] = x2cos(1+x3)

    Integral x2cos(1+x3)dx

    I got (x3 /3)(-sin(1+x3)(3x2) ... I have no idea how that turns out to be what it is....

    c) "Find the derivative and state a corresponding integration formula."

    (d/dx)[sqrt(x3+5)] = (3x2)/(2)[sqrt(x3​+5)]

    Integral (3x2)/(2)[sqrt(x3​+5)] dx

    Once again... I understand how to take the integral of [sqrt(x3​+5)]-1/2 .... but I don't understand what happens to (3/2)x2 ...?


    Thank you!
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  2. #2
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    Re: A few integration problems

    Tomatoes.........

    The integration process is the reverse of differentiation.

    since (X)(1+X2)-1/2 is the derivateive of your function f(x) = sqr(1+x^2) then as it is in the integration formula it will produce the antiderivative ..i.e the function from which (X)(1+X2)-1/2 came out as derivative.....this is what we call indefinite integral or antiderivative.......the dx which accompanies the integral form is the so called differential of the variable x and it is compalsory to put it there....why? it is difficult to explain...
    have a look here to understand a little bit about the differentials....

    Differential of a function - Wikipedia, the free encyclopedia

    Actually we integrate differentials........
    MINOAS
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  3. #3
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    Re: A few integration problems

    Quote Originally Posted by tomatoes View Post
    "In each part, confirm that the formula is correct."
    This is what you haven't done. Instead of seeing why this...



    ... might be true, if it is true, you have decided to go right ahead and do this...



    Now, you say...

    Quote Originally Posted by tomatoes View Post
    I understand how to take the integral of (1+X2)-1/2 ,
    ... but are you sure? Do you mean this...



    ... ? If you know that, somehow, then, good. But it's true because of this...



    ... and explaining that is harder than what's on the menu here. (But see the bottom spoiler, below.)

    So do as suggested and confirm the given derivatives. Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    The general drift is...



    or, integrating...




    Similarly, you are invited to see that...



    ... is basically the same whether you read it from top down (differentiating) or bottom up (integrating). And you should use the chain rule to confirm that it's true...



    Try doing the same for c):



    Good luck.

    That derivative of arsinh:

    Spoiler:


    Hope that helps. More orthodox advice might be forthcoming too, no doubt. But the exercise is designed to make you confront the chain rule, one way or another.

    Btw, it's best to read dx as just 'with respect to x', e.g. \dfrac{d}{dx} means 'the derivative with respect to x of...' and \int\ y\ dx means 'the integral with respect to x of...' (... of y or whatever).

    Cheers.

    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

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