Integration distributes over addition and subtraction but never over multiplication. Integrating each factor of a product and multiplying the results is the single most common mistake in integration - here is why it fails, and what to do instead.
Integration splits neatly over addition and subtraction, so it feels as though it should split over multiplication in the same way. It does not. Integrating each factor of a product on its own and multiplying the results is the single most common mistake in integration.
Integration passes term by term through a sum or a difference:
Take , whose integral is . Break it up across a plus sign and integrate the pieces:
Split the function however you like across and , integrate each part, and the answer always comes back correct.
There is no matching rule for a product. Writing one down is the mistake:
Integrate , first the wrong way, then the right way.
The two answers are not a small slip apart — they are different functions, and only differentiates back to .
If the product simplifies, simplify it first and integrate once. Collecting into is the whole job; the power rule then finishes it in one step.
If it will not simplify — a genuine function times a function, such as — there is a dedicated technique called integration by parts, covered in a later lesson. Multiplying two separate integrals is never the way out.