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Math Help - Evaluate the definite integrals

  1. #1
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    Evaluate the definite integrals

    I have absolutely no idea how I'm supposed to do this.

    Evaluate each of the following definite integrals:

    a) \int_1^4 \frac{x^2+2x}{\sqrt{x^2}} dx
    b) \int_{-1}^{1} \frac{1}{v}(3v^3-v^\frac{-1}{3}) dv

    Can anyone point me in the right direction? I've done definite integrals, but none that hard before.

    Thanks in advance
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  2. #2
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    Worked out how to do it.

    You integrate it first, then multiply it by the bigger number and minus it by the smaller number times the value.
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  3. #3
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    The easiest way is to simplify the integrand...

    In the region \displaystyle 1 \leq x \leq 4, \sqrt{x^2} = x.

    So \displaystyle \int_1^4{\frac{x^2+2x}{\sqrt{x^2}}\,dx} = \int_1^4{\frac{x^2+2x}{x}\,dx} = \int_1^4{x + 2\,dx}.


    Also \displaystyle \int_{-1}^1{\frac{1}{v}(3v^3 - v^{-\frac{1}{3}})\,dv} = \int_{-1}^1{3v^2 - v^{-\frac{4}{3}}\,dv}.
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