IB DP Maths: AA HL复习笔记5.9.1 Integrating Further Functions

As with other problems in integration the results in this revision note may have further uses such as

• evaluating a definite integral
• finding the constant of integration
• finding areas under a curve, between a line and a curve or between two curves

cosec (cosecant, csc), sec (secant) and cot (cotangent) are the reciprocal functions of sine, cosine and tangent respectively.

What are the antiderivatives involving reciprocal trigonometric functions?

• These are not given in the formula booklet directly
  • they are listed the other way round as ‘standard derivatives’
  • be careful with the negatives in the last two results
  • and remember “+c” !

How do I integrate these if a linear function of x is involved?

• All integration rules could apply alongside the results above
• The use of reverse chain rule is particularly common
  • For linear functions the following results can be usefulThese are not in the formula booklet
  • • they can be deduced by spotting reverse chain rule
    • they are not essential to remember but can make problems easier

arcsin, arccos and arctan are (one-to-one) functions defined as the inverse functions of sine, cosine and tangent respectively.

What are the antiderivatives involving the inverse trigonometric functions?

How do I integrate these expressions if the denominator is not in the correct form?

What are the antiderivatives of exponential and logarithmic functions?

• How do I integrate exponentials and logarithms with a linear function of x involved?
• These are not in the formula booklet but can be derived using ‘adjust and compensate’ from reverse chain rule

• Exam Tip