# Integrating Trig Functions

# Integrating Trig Functions

**Integrating Trig Functions**

**Integrating** **trigonometric functions** is little more than both an exercise in **memory** and **application** of that which we have already learned. It **combines all of the skills so far** and allows for **very difficult-looking functions** to be **integrated**.

Make sure you are happy with the following topics before continuing.

**Some Results Have to be Memorised**

There is **no easy way to derive** these results, so the best strategy is to **commit them to memory**.

\int \sin(x)dx=-\cos(x)+c

\int \cos(x)dx=\sin(x)+c

\int \sec^{2}(x)dx=\tan(x)+c

\int \cosec(x)\cot(x)dx=-\cosec(x)+c

\int \sec(x)\tan(x)dx=\sec(x)+c

\int \cosec^{2}(x)dx=-\cot(x)+c

If there is a **coefficient** before x, then you **divide by the coefficient** when you** integrate** (just like when **integrating** **exponentials**).

\int \sin(ax+b)dx=-\dfrac{1}{a}\cos(ax+b)+c

The other results follow in a **similar way**.

**Other Results Can Be Derived**

Other results can be **derived** using methods we have **already learned**.

**Example: **Find the **integral** of \tan(x).

A Level

**Double Angle Formulas and Identities**

We cannot** integrate** functions such as \sin^{2}x directly, but we can **integrate** functions like \sin(2x). This means that we can **rearrange the double angle formulas** to be able to **integrate** many more **trigonometric** functions. These are the **most important ones** to **remember**.

\sin^{2}(x)=\dfrac{1}{2}(1-\cos(2x))

\cos^{2}(x)=\dfrac{1}{2}(1+\cos(2x))

\sin(x)\cos(x)=\dfrac{1}{2}\sin(2x)

\dfrac{2\tan(x)}{1-\tan^{2}(x)}=\tan(2x)

We can use** identities** for the **same purpose**. For example, we cannot** integrate** \tan^{2}(x) directly, but we can **integrate** \sec^{2}(x), so the **identities** below can help.

\tan^{2}(x)=\sec^{2}(x)-1

\cot^{2}(x)=\cosec^{2}(x)-1

A Level**Example 1: Using Results from Memory**

**Integrate** \sec^{2}(3x)

**[1 mark]**

The result from **memory** is:

Now add in the **coefficient** of x:

A Level

**Example 2: Double Angle Formulas**

**Integrate** \sin(x)(\sin(x)+\cos(x))

**[3 marks]**

A Level

## Example Questions

**Question 1: **Integrate:

i) \cos(3x)

ii) \sin(4x+1)

iii) \cosec(3-x)\cot(3-x)

iv) \sec^{2}(199x)

v) \cosec^{2}\left(\dfrac{1}{3}x+\dfrac{2}{7}\right)

vi) \sec\left(\dfrac{1}{9}x\right)\tan\left(\dfrac{1}{9}x\right)

**[6 marks]**

i) \int \cos(3x)dx=\dfrac{1}{3}\sin(3x)+c

ii) \int\sin(4x+1)dx=-\dfrac{1}{4}\cos(4x+1)+c

iii) \int \cosec(3-x)\cot(3-x)dx=\cosec(3-x)+c

iv) \int \sec^{2}(199x)dx=\dfrac{1}{199}\tan(199x)+c

v) \int \cosec^{2}\left(\dfrac{1}{3}x+\dfrac{2}{7}\right)dx=-3\cot\left(\dfrac{1}{3}x+\dfrac{2}{7}\right)+c

vi) \int \sec\left(\dfrac{1}{9}x\right)\tan\left(\dfrac{1}{9}x\right)dx=9\sec\left(\dfrac{1}{9}x\right)+c

**Question 2: **What is the integral of \cot(x)?

**[2 marks]**

**Question 3: **Integrate \sin^{2}(x)-\cos^{2}(x)

**[3 marks]**

**Question 4: **Find the integral of 3\cot^{2}(4x)

**[3 marks]**

## Worksheet and Example Questions

### Integration

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