# Integration By Substitution

The counterpart of the chain rule for integration. If the variable $$x$$ over which a function $$f$$ is integrated with is substituted with a function $$g(y)$$ then
$\int_{g(a)}^{g(b)} f(x) \mathrm{d}x = \int_a^b f(g(y)) g^\prime(y)\mathrm{d}y$
or, equivalently
$\int_{a}^{b} f(x) \mathrm{d}x = \int_{g^{-1}(a)}^{g^{-1}(b)} f(g(y)) g^\prime(y)\mathrm{d}y$
For functions of more than one argument the transformation is given by
$\int_{\mathbf{g}(\mathbf{C})} f(\mathbf{x}) \mathrm{d}x_0 \mathrm{d}x_1 \dots \mathrm{d}x_n = \int_{\mathbf{C}} f(\mathbf{g}(\mathbf{y})) \Biggl|\left|\tfrac{\partial\left(x_0,x_1,\dots,x_n\right)}{\partial\left(y_0,y_1,\dots,y_n\right)}\right|\Biggr| \mathrm{d}y_0 \mathrm{d}y_1 \dots \mathrm{d}y_n$
where $$\left|\frac{\partial\left(x_0,x_1,\dots,x_n\right)}{\partial\left(y_0,y_1,\dots,y_n\right)}\right|$$ is the the determinant of the Jacobian of $$\mathbf{g}$$ and the outer vertical bars represent the absolute value of the expression between them.

### Gallimaufry

 AKCalc ECMA Endarkenment Turning Sixteen

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