Reverse the standard derivatives: ∫ eᵏˣ dx = (1/k)eᵏˣ + c, ∫ cos kx dx = (1/k) sin kx + c, ∫ sin kx dx = −(1/k) cos kx + c and ∫ 1/x dx = ln|x| + c.
For (ax + b)ⁿ, raise the power by one and divide by the new power and by a: ∫ (5x + 1)³ dx = (5x + 1)⁴/20 + c. Always check by differentiating, using the chain rule.
Worked example
Find ∫ 5 sin 3x dx.
- ∫ sin 3x dx = −(1/3) cos 3x.
- Multiply by 5.
Answer:−(5/3) cos 3x + c
Key words
- ln|x|
- the natural log of the size of x
- reverse chain rule
- divide by the inner derivative
- standard integral
- a result worth knowing by heart
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