Integration — FAQ

An indefinite integral, $\int f(x) dx$, gives a function (or family of functions, $F(x)+C$) representing the antiderivative. A definite integral, $\int_a^b f(x) dx$, gives a single numerical value representing the signed area under the curve between two points.

For indefinite integrals, it is critically important and you will lose a mark if you forget it. It represents an entire family of functions that have the same derivative. For definite integrals, the '+ C' cancels out when you evaluate $F(b) - F(a)$, so it's not needed in the final calculation, but the underlying indefinite integral technically still has it.

Use substitution when the integrand contains a function and its derivative (or something close to it), often in a 'function-of-a-function' or composite form. Use integration by parts when the integrand is a product of two unrelated functions, like $x e^x$ or $x^2 \sin x$.

The original limits are $x$-values. When you change the entire integral to be in terms of a new variable, $u$, the limits must also be expressed in terms of $u$. This allows you to complete the calculation without ever having to substitute back to $x$, which is more efficient and less error-prone.