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March 30, 2017

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Using an integration formula,what is the indefinite integral of (sign for integral)(cos(4x)+2x^2)(sin(4x)-x)dx. Any help very much appreciated.

  • calculus - ,

    An interesting problem! In fact, it would make a good exam question, because you need to know four rules to calculate the integral, or equivalent to four short questions.

    (cos(4x)+2x^2)(sin(4x)-x)
    =cos(4x)sin(4x)-xcos(4x)+2x²sin(4x)-2x³

    1. cos(4x)sin(4x)
    use substitution
    u=sin(4x)
    du=4cos(4x)dx
    so
    ∫cos(4x)sin(4x)dx
    =(1/4)∫udu (use the power rule)

    2. -xcos(4x)
    Use integration by parts:
    ∫-xcos(4x)dx
    =-(1/4)xsin(4x)+(1/4)∫sin(4x)

    3. 2x²sin(4x)
    Use integration by parts, similar to (2) above.

    4. -2x³
    use the power rule.

    Post your answer for a check if you wish. Do not forget the constant of integration (C).

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