New Zealand
Level 8 - NCEA Level 3

# Integration of Sine and Cosine

Lesson

We discovered that a special relationship exists when differentiating sine and cosine functions.

We saw that $\frac{d}{dx}\sin\left(x\right)=\cos\left(x\right)$ddxsin(x)=cos(x) and $\frac{d}{dx}\cos\left(x\right)=-\sin\left(x\right)$ddxcos(x)=sin(x)

Since integration reverses the process of differentiation, we can deduce that:

#### Worked Examples

##### example 1

State a primitive function of $6\sin x-\cos x$6sinxcosx.

You may use $C$C as a constant.

##### Example 2

Integrate $9\sin3x$9sin3x.

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

We know from our experience with differentiation, that the derivative of $\cos3x$cos3x is $-3\sin3x$3sin3x. So we have this factor of $3$3 in our expression.

Ignoring the $9$9 in our question, if we were just looking at integrating $\sin3x$sin3x, then we need to remove this factor of $3$3 by dividing through by $3$3

Putting it all together we have:

In general then:

##### Example 3

Integrate $-5\cos\left(\frac{x}{4}\right)$5cos(x4).

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

##### example 4

Evaluate $\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(4\cos x+\cos4x\right)dx$π6π6(4cosx+cos4x)dx.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91579

Apply integration methods in solving problems