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.