[TIL] Rationalizing Substitution

Rationalizing Substitution is a special type of U-Substitution, radicals in function often cause problems when integrating, use rationalizing substitution we can transform the problem to help us solve them.

For example, sometimes our function contain something like $$\sqrt{x}$$, in order to eliminate the “square root”, we can make substitution $$x = z^2$$，$$dx = 2z dz$$. Then you may can solve function’s integration easier.

U-Substitution

Today I learned U-Substitution, it is a method for finding integrals in calculus, also known as integration by substitution. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, U-Substitution is an important tool in mathematics. It is the counterpart to the chain rule of differentiation.

If you find difficult to finding an antiderivative directly, sometimes you may want use substitution $$u = ϕ(x)$$, rewrite $$f(x)dx$$ form to some function $$g(u)du$$ form and get antiderivative of $$g(u)du$$ first, and put $$x$$ in the antiderivative to get what you want at first.

For example, consider the integral

$$\int _{0}^{2}x\cos(x^{2}+1)\,dx$$

If we apply the formula from right to left and make the substitution $$u = ϕ(x) = (x2 + 1)$$, we obtain $$du = 2x dx$$ and hence; $$x dx = ½du$$

{\begin{aligned}\int _{x=0}^{x=2}x\cos(x^{2}+1)\,dx&{}={\frac {1}{2}}\int _{u=1}^{u=5}\cos(u)\,du {}={\frac {1}{2}}(\sin(5)-\sin(1)).\end{aligned}}

see also: Integration by substitution