Integration by substitution

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TODO: Flesh this page out with explanation and Jacobian reference

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1 dimensional example

Question:

$$\int_0^3\frac{4x}{(1+x^2)^2}dx$$

That $1+x^2$ is the problem, and the derivative is clearly something to do with just $x$ (infact $=2x$) so this invites a substitution.

• Substitute $$u=1+x^2$$ then $$\frac{du}{dx}=2x$$ thus $$du=2x dx$$

• • Notice our integral is $$I=\int_0^3\frac{4x}{(1+x^2)^2}dx=\int_0^3\frac{2}{(1+x^2)^2}2xdx$$ we can literally replace the $2xdx$ with $du$ - this is a nice shortcut.
  • More formally we actually have $dx=\frac{du}{2x}$ let us substitute this

    $$\int_0^3\frac{4x}{(1+x^2)^2}dx=\int^{x=3}_{x=0}\frac{4x}{u^2}\frac{du}{2x}$$

    Notice: I write explicitly the limits of the integral because I have yet to express them in terms of $u$

    $$=\int^{x=3}_{x=0}\frac{2}{u^2}du=2\int^{x=3}_{x=0}\frac{1}{u^2}du$$

    • Converting the limits:
      • $x=3\implies u=1+3^2=10$
      • $x=0\implies u=1$

    • Thus $$I=2\int^{10}_{1}\frac{1}{u^2}du$$ which the reader ought to be able to integrate.