Reversing the Chain Rule
An indefinite integral is the family of all antiderivatives, and u-substitution is the chain rule read backwards.
Antiderivatives
The indefinite integral of f is any function whose derivative is f, written with a C because derivatives cannot see constants. If F works, then F plus any constant also works, so the answer is a family, not a single function. Basic rules mirror differentiation: powers, exponentials, and sines each reverse cleanly.
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\quad n \neq -1The Substitution Idea
When an integrand contains a function and (a multiple of) its derivative, the chain rule is hiding inside. Pick u to be the inner function so that du covers the leftover factor. The integral then collapses to a basic form in u, you integrate, and substitute back at the end.
\int f(g(x))g'(x)\,dx = \int f(u)\,duA Worked Swap
Take the integral of x times the square root of (1 + x^2). Let u = 1 + x^2, so du = 2x dx and the x dx becomes du/2. The integral is now one half the integral of u to the one half power, which gives u to the three halves over three, so the final answer is (1 + x^2) to the three halves over 3 plus C.
\int x\sqrt{1+x^2}\,dx = \frac{(1+x^2)^{3/2}}{3} + CFlashcard
Constant of integration
Check your understanding
To evaluate the integral of 2x times e to the (x squared), what substitution fits best?