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Integrating Special Functions (DP IB Maths: AI HL)
Revision Note
Integrating Trig Functions
How do I integrate sin, cos and 1/cos2?
- The antiderivatives for sine and cosine are
where is the constant of integration
- Also, from the derivative of
- All three of these standard integrals are in the formula booklet
- For the linear function, where and are constants,
- For calculus with trigonometric functions angles must be measured in radians
- Ensure you know how to change the angle mode on your GDC
Examiner Tip
- Make sure you have a copy of the formula booklet during revision but don't try to remember everything in the formula booklet
- However, do be familiar with the layout of the formula booklet
- You’ll be able to quickly locate whatever you are after
- You do not want to be searching every line of every page!
- For formulae you think you have remembered, use the booklet to double-check
- However, do be familiar with the layout of the formula booklet
Worked example
a)
Find, in the form, an expression for each integral
b) A curve has equation.
The curve passes through the point with coordinates.
Find an expression for.
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Integrating e^x & 1/x
How do I integrate exponentials and 1/x?
- The antiderivatives involving and are
where is the constant of integration
-
- These are given in the formula booklet
- For the linear function, where and are constants,
- It follows from the last result that
-
- which can be deduced using Reverse Chain Rule
- With ln, it can be useful to write the constant of integration,, as a logarithm
- using the laws of logarithms, the answer can be written as a single term
- where is a constant
- This is similar to the special case of differentiating when
Examiner Tip
- When revising, familiarise yourself with the layout of this section of the formula booklet, make sure you know what is and isn't in there and how to find it very quickly
Worked example
A curve has the gradient function.
Given the exact value of is find an expression for.
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