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Standard Level

13.03 Antidiffferentiation and trigonometric functions

Lesson

Recall that we previously established the following results when differentiating trigonometric functions:

Differentiating trigonometric functions
$$ddxsinx $$= $$cosx
$$ddxcosx $$= $$sinx


and

$$ddxsin(ax+b) $$= $$acos(ax+b)
$$ddxcos(ax+b) $$= $$asin(ax+b)

 

Reversing these we get the following rules for integrating trigonometric functions:

Integration of trigonometric functions
$$cosxdx $$= $$sinx+C
$$sinxdx $$= $$cosx+C


and

$$cos(ax+b)dx $$= $$1asin(ax+b)+C
$$sin(ax+b)dx $$= $$1acos(ax+b)+C


Where $$a, $$b and $$C in each case are constants and $$a0

 

Worked examples

Example 1

Determine $$5cos(2x+π3)dx.

Think: To integrate we are going to divide by $$a from the term $$ax+b; here $$a=2. Then we change the function from cosine to sine.

Do:

$$5cos(2x+π3)dx=52sin(2x+π3)+C, where $$C is a constant.


Example 2

If $$f(x)=0.5sin(x4) and $$f(2π)=1, find $$f(x).

Think: We can first find the indefinite integral, and then use the given point $$(2π,1) to find the value of the constant of integration.

For our integral, we will use the rule $$sin(ax+b)dx=1acos(ax+b)+C. We have $$a=14, we want to divide by $$a, as well as change the sign and function. Remember, dividing by $$14 is the same as multiplying by $$4.

Do:

$$0.5sin(x4)dx $$= $$4×0.5cos(x4)+C

Multiply by $$4, and change the function and sign

  $$= $$2cos(x4)+C, where $$C is a constant

Simplify

 

Using the point $$(2π,1), find $$C:

$$f(2π) $$= $$1
$$2cos(2π4)+C $$= $$1
$$0+C $$= $$1
$$C $$= $$1

Thus, $$f(x)=2cos(x4)1.

 

Practice questions

Question 1

Integrate $$5cos(x4).

You may use $$C as the constant of integration.

Question 2

State a primitive function of $$6sinxcosx.

You may use $$C as a constant.

Question 3

Given that $$f(x)=kcos3x, for some constant $$k, and that $$f(0)=2 and $$f(π6)=6:

  1. Determine the value of $$k.

  2. Now find an expression for $$f(x)

    You may use $$C to represent an unknown constant.

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