Standard integrals
Reverse the standard derivatives: cos x integrates to sin x, sin x to -cos x, e^x to e^x and 1/x to ln|x|, each plus C.
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Analogy
A reverse times table, memorised so you can recognise patterns quickly.
Example
Antiderivative: -cos x.At pi: -cos(pi) = 1.At 0: -cos(0) = -1.Result: 1 - (-1) = 2.
In practice
Every harder technique ends by reducing the problem to one of these.
Practice
Flashcards
- What is the integral of sin x from 0 to pi?2.
- What is the integral of 1/x dx?ln|x| + C.
Questions people ask
Why does the integral of sin x have a minus sign?
Because the derivative of cos x is minus sin x. To get plus sin x back, you need minus cos x. You can always check by differentiating your answer: the derivative of minus cos x is sin x, which matches the original.
Why is the integral of 1/x written with absolute value signs?
The natural logarithm only accepts positive inputs, but 1/x exists for negative x too. Writing ln of absolute x covers both sides of zero, and its derivative is still 1/x. Over any interval that crosses zero, the integral does not exist.