U-substitution turns a messy integral into a simple one by renaming the inside
Faced with the integral of (2x cubed plus 1) to the seventh power times x squared, expanding the bracket would be miserable. Call the inner expression u instead, notice its derivative is already lurking nearby, and the whole thing collapses to u to the seventh. That is the chain rule run in reverse.
Also called the reverse chain rule or change of variables, substitution evaluates integrals and antiderivatives by spotting a composite function multiplied by the derivative of its inner part. In the opening example, the inner expression differentiates to 6x squared, so after pulling out a factor of one sixth the integrand becomes u to the seventh, du. Integrating gives u to the eighth over 48; swapping the original bracket back in for u, and adding a constant, finishes the job. Differentiating that answer recovers the original integrand, a check worth doing every time, since not every integral yields to the method.
For definite integrals the limits move too. If g has a continuous derivative, integrating f(g(x)) times g-prime(x) from a to b equals integrating f(u) from g(a) to g(b). Proving it takes the chain rule plus the fundamental theorem of calculus applied twice. Heuristically, treating du and dx as infinitesimals suggests the formula, and differential forms make that manipulation rigorous, which partly vindicates Leibniz's notation.
Classic uses abound. With u equal to x squared plus 1, the integral of x times the cosine of that expression becomes half the sine of it. Rewriting tangent as sine over cosine and letting u be cosine gives the natural log of the absolute secant, and cotangent works the same way with sine. For x over the square root of x squared plus 1, integrated from 0 to 2, the new limits run from 1 to 5 and the answer is the square root of 5 minus 1.
The formula can be read in either direction. Left to right, it introduces a new variable for an inner expression. Right to left, it replaces the original variable with a function of a new one, the basis of trigonometric substitution, used for instance on the square root of 1 minus x squared from 0 to 1.
Source: Integration by substitution
