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Thread: U substitution

  1. #1
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    U substitution

    Quick question about u substitution. When do you know when you need to change coordinates? I remembered learning about this in my second course of calculus, and now i have to revisit it, but i forgot how it works. A example with a brief explanation would be appreciated, or any external inks as well. Thanks.
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  2. #2
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    I was always told to change the terminals when the variable itself changed.

    Consider the following problem

    $\displaystyle \int_2^3\frac{2x}{x^2+1}~dx$

    We can see that $\displaystyle 2x$ is the derivative of $\displaystyle x^2+1$ therefore we make

    $\displaystyle u = x^2+1 \Rightarrow \frac{du}{dx} = 2x$

    As the terminals x = 2,3 are with respect to x and our substituted variable is u we use

    $\displaystyle u = x^2+1 $ to change them.

    x=2 gives $\displaystyle u = 2^2+1 = 5$

    x=3 gives $\displaystyle u = 3^2+1 = 10$

    And now we have

    $\displaystyle \int_5^{10}\frac{1}{u}\frac{du}{dx}~dx$

    $\displaystyle \int_5^{10}\frac{1}{u}~du$

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