Many antiderivatives are not obtained by a single basic formula. Integration techniques work by recognizing the derivative rule that would produce the integrand and reversing it.

Substitution

Substitution reverses the chain rule. If an integrand contains a function together with its derivative, set

$$u=g(x),\qquad du=g'(x),dx.$$

For example,

$$\int 2x\cos(x^2),dx$$

becomes

$$\int \cos u,du=\sin u+C,$$

so

$$\int 2x\cos(x^2),dx=\sin(x^2)+C.$$

Integration by parts

The product rule leads to

$$\int u,dv=uv-\int v,du.$$

For example,

$$\int x e^x,dx =xe^x-\int e^x,dx =xe^x-e^x+C.$$

Definite integrals

A substitution can be used in a definite integral either by converting the bounds to the new variable or by returning to the original variable before applying the bounds.

Integration by parts also applies directly with bounds:

$$\int_a^b u,dv=\left[uv\right]_a^b-\int_a^b v,du.$$

The goal is not to memorize isolated tricks, but to identify structure in the integrand and choose a transformation that reduces it to known antiderivatives.