Spotting the inner function, the factor that has to already be there, and a check that makes you right every time.
Every integral here is a chain rule someone already did. The job is to work out what it was. Pick an integral and check your answer by differentiating it back.
∫ sin(2x + 5) dx
= −½cos(2x + 5) + c
Now the check. The left number is the thing you were integrating. The right number is what the answer gives when you differentiate it back. If the answer is right they agree at every x, so move the slider and try to break it.
The derivative is worked out numerically from a tiny gap, so it is a genuine check and not a restatement.
∫ f′(g(x)) · g′(x) dx = f(g(x)) + c. In words: if the derivative of the inside is already sitting there as a factor, you can integrate the outside and leave the inside alone.
So the question to ask every time is: what is the inside, and is its derivative present?
A constant you can fix. A variable you cannot. If the inside is linear, like 2x + 5, its derivative is a number and you simply divide by it. If the inside is x² you need an x already present as a factor; you cannot conjure one, and no amount of dividing will rescue ∫sin(x²)dx, which is not something you are asked to do.
At Standard Level the power n could not be −1. Here it can: ∫ 1x dx = ln|x| + c. That is why 1⁄(3x + 2) integrates to a logarithm rather than a power, and it is the case most often missed by anyone still reaching for "add one to the power".
| ∫ f(x) dx | sin x | cos x | ex | 1cos²x | 1x |
|---|---|---|---|---|---|
| gives | −cos x | sin x | ex | tan x | ln|x| |
1. ∫ cos(3x) dx is:
2. Which of these can be integrated by inspection?
3. Evaluate ∫₀¹ e2x dx to 3 decimal places.
Identifying the inner function is the method mark. Writing u = 3x + 2 and du = 3dx makes it explicit and is never wasted, even when you can see the answer.
The dividing constant is where most answers go wrong. A quick differentiate back, mentally, catches it in seconds.
Keep the modulus in ln|…|. It costs nothing and it is sometimes an explicit requirement.
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