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    Hello there,

    Could someone please tell me how would I integrate 6sin(7y+5) please. I understand for example that 4cos3y when integrated will equal 4/3sin3y but I am unsure on what to do with the bracket.

    Thank you. :P
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    dy or dx
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    (Original post by ncsjohn02)
    dy or dx
    What happens to the (7y+5) part sorry?
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    (Original post by apo1324)
    Hello there,
    I understand for example that 4cos3y when integrated will equal 4/3sin3y but I am unsure on what to do with the bracket.
    This is correct but why do you understand it?

    Have you done integration by substitution? Letting u=3y and following the process will show you why it integrates to (4/3)sin3y.

    For your problem, try substituting u=7y+5 and see what happens.

    The rules of integration are not always opposite to differentiation so it's not a good idea to just learn integrals without knowing how they're derived.
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    the answer is -6/7cos(7y+5) i think. you just need to multiply your answer by 1/the differential of the bracket
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    (Original post by jimmyreed112)
    the answer is -6/7cos(7y+5) i think. you just need to multiply your answer by 1/the differential of the bracket
    Oh ok, but the answer is -6sin(7y+5)... I don't understand where the minus sign came from.
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    jimmyreed112 is correct for this example, although this will not work for polynomials above degree 1 in the bracket. I'm not entirely sure that if you had a polynomial above degree 1 that it can be integrated with elementary functions at all...
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    (Original post by notnek)
    This is correct but why do you understand it?

    Have you done integration by substitution? Letting u=3y and following the process will show you why it integrates to (4/3)sin3y.

    For your problem, try substituting u=7y+5 and see what happens.

    The rules of integration are not always opposite to differentiation so it's not a good idea to just learn integrals without knowing how they're derived.
    No, I haven't done integration by substitution. Yeah, basically my teacher gave us a list of all the integrals and expects us to learn them.
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    The minus sign comes from the fact that the integral of sin is -cos, that's all.
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    the integral of sinx is -cosx...its just a case of learning the rules of integrating and differentiating trigonometric functions
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    For this example... -2cos(10-x):-

    Would it be -2sin(10-x) + c as although 10-x differentiated = -1 cos becomes a negative again when integrating sin?? Thanks for the help everyone, appreciate it btw. so -2/1sin(10-x) + c.
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    the integral of -2cos(10-x) is 2sin(10-x) + c and yes youre right
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    (Original post by jimmyreed112)
    the integral of -2cos(10-x) is 2sin(10-x) + c and yes youre right
    Thanks.
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    (Original post by apo1324)
    Thanks.
    anytime
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    If you want to do it another (and safer) way, use the fact that sine integrates to some multiple of cosine, and cosine integrates to some multiple of sine (when what's in the brackets is linear, but in C3 it will always be linear).

    So you can let \displaystyle \int \sin(ax+b)\, dx = K\cos(ax+b) + C. Then differentiate the RHS and compare it to \sin (ax+b) to find the value of K.

    So for example let's say we have to integrate \cos (3-11x). Then we can say:
    \displaystyle \int \cos(3-11x)\, dx = K\sin (3-11x) + C

    Differentiating the both sides gives
    \cos (3-11x) = -11K\cos(3-11x)

    ...and so we must have -11K=1, and hence K=-\dfrac{1}{11}, so our final answer is:

    \boxed{ \displaystyle \int \cos(3-11x)\, dx = -\dfrac{1}{11}\sin(3-11x) + C }

    Hope this helps.

    [EDIT: Thanks ghostwalker for the correction]
 
 
 
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