Constant rule
Example. 1
Theorem. 1 (Constant rule)
Proof:
Polynomial
Theorem. 2 (Addition of integrands)
Proof:
Integrate both sides once gives the desired result.
Theorem. 3 (Power rule)
Proof: By power rule in the differentiation version:
Why does it fail at n = -1? Power rule in differentation failed in n = 0 because "this is not a polynomial, instead this is a constant so it does not work". Now for n = -1, the denominator becomes zero, making the integration result failed. However, it is clear that 1/x has an anti-derivative since it is almost everywhere continuous...
Theorem. 4 (Exception of power rule)
Proof: omitted.
Note: 1/x diverge at zero, that makes a bit a technical problem but we are not going to discuss it here. It can be solved by p.v. integration but it is far from our scope.
The trigonometrical cycle
In differentiation we know that sin x -> cos x -> - sin x -> - cos x -> sin x is a differentiation cycle, i.e.
Since integration is the inverse of differentiation, integration actually reverses the cycle.
Under integration, the cycle goes like sin x -> - cos x -> - sin x -> cos x -> sin x. It is exactly the reversed version of differentiation cycle of trigonometric functions. Readers should realize the fact that integration is finding the anti-derivative of a function, hence it must be the reversed cycle of the differentiation ones.
Practical example
In this section we know nothing about chain rule, substituion or other technique yet, so the main point is that you should identify and separate each term and integrate each of them by power rule or trigonometrical cycle.
Example. 2 Evaluate
Solution:
Approach: you may notice that
Example. 3 Evaluate
Solution: separate each term gives
Apply power rule and trigonometric cycle yields
Again note that in the final answer only one constant left.
Such skills are very basic and its difficulty grows like geometric series in later stage, so basic skills should be treated seriously and mass exercises are adviced.
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