Constant and Sum Rules for Integration

Constant and Sum Rules for Integration

When integrating, you can handle constants and sums term by term. These rules make many expressions much easier to work with.

1) Use the constant rule

A constant factor can be pulled outside the integral: [ \int c,f(x),dx = c\int f(x),dx ] This means you do not need to integrate the constant separately.

2) Use the sum rule

Integrals distribute over addition and subtraction: [ \int \big(f(x)+g(x)\big),dx = \int f(x),dx + \int g(x),dx ] The same idea works for subtraction.

3) Integrate each part

After rewriting the expression, integrate each term using basic antiderivatives. For example, powers of (x), constants, and simple functions can often be integrated one at a time.

4) Simplify the result

Combine like terms and write the constant of integration (+C) when the integral is indefinite.

Quick check

Differentiate your final answer. If you recover the original integrand, your integration is correct.

Example structure

If you see something like a constant times a sum, first factor out the constant, then split the integral, then integrate each term separately.