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Taking the derivative of (x2+5)4(x^2 + 5)^4 takes seconds using the chain rule, but how do you reverse the process to integrate 2x(x2+5)3 dx2x(x^2 + 5)^3 \, dx without expanding the polynomial?
In 1675, Gottfried Wilhelm Leibniz introduced differential notation (dxdx) specifically to make chain rule transformations behave just like algebraic fraction substitutions.
How do we systematically trade a messy composite integral for a basic textbook antiderivative?

The Substitution Recipe

The u-substitution method simplifies an integral by defining a new variable u=g(x)u = g(x), whose differential is du=g′(x) dxdu = g'(x) \, dx.
Let's evaluate ∫2xcos⁡(x2) dx\int 2x \cos(x^2) \, dx. We set the inner function to u=x2u = x^2, which makes its derivative differential du=2x dxdu = 2x \, dx.
Replacing x2x^2 with uu and 2x dx2x \, dx with dudu transforms the expression into ∫cos⁡(u) du=sin⁡(u)+C\int \cos(u) \, du = \sin(u) + C, which back-substitutes to sin⁡(x2)+C\sin(x^2) + C.
What happens when the derivative factor in your integral is off by a constant multiplier?

Balancing Constant Multipliers

Because integrals are linear, any constant factor can move freely across the integral sign, whereas variables can never leave the integrand.
Consider ∫x2x3+5 dx\int x^2 \sqrt{x^3 + 5} \, dx. Choosing u=x3+5u = x^3 + 5 gives du=3x2 dxdu = 3x^2 \, dx, meaning x2 dx=13 dux^2 \, dx = \frac{1}{3} \, du.