Integrating Special Functions (DP IB Maths: AA HL)

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Paul

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Integrating Trig Functions

How do I integrate sin, cos and sec^2?

The antiderivatives for sine and cosine are

$\int \sin x d x = - \cos x + c$

$\int \cos x d x = \sin x + c$

where $c$ is the constant of integration

Also, from the derivative of $\tan x$

$\int \sec^{2} x d x = \tan x + c$

The derivatives of $\sin x, \cos x$ and $\tan x$ are in the formula booklet
- so these antiderivatives can be easily deduced
For the linear function $a x + b$ , where $a$ and $b$ are constants,

$\int \sin (a x + b) d x = - \frac{1}{a} \cos (a x + b) + c$

$\int \cos (a x + b) d x = \frac{1}{a} \sin (a x + b) + c$

$\int \sec^{2} (a x + b) d x = \frac{1}{a} \tan (a x + b) + c$

For calculus with trigonometric functions angles must be measured in radians
- Ensure you know how to change the angle mode on your GDC

Exam Tip

The formula booklet can be used to find antiderivatives from the derivatives
- Make sure you have the page with the section of standard derivatives open
- Use these backwards to find any antiderivatives you need
- Remember to add 'c', the constant of integration, for any indefinite integrals

Worked example

Find, in the form

F (x) + c

, an expression for each integral

$\int \cos x d x$
$\int \sec^{2} (3 x - \frac{π}{3}) d x$

5-4-1-ib-hl-ai-aa-extraaa-we1a-soltn

b) A curve has equation

y = \int 2 \sin (2 x + \frac{π}{6}) d x

The curve passes through the point with coordinates

(\frac{π}{3}, \sqrt{3})

Find an expression for

y

5-4-1-ib-hl-ai-aa-extraaa-we1b-soltn-

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Integrating e^x & 1/x

How do I integrate exponentials and 1/x?

The antiderivatives involving $e^{x}$ and $\ln x$ are

$\int e^{x} d x = e^{x} + c$

$\int \frac{1}{x} d x = \ln | x | + c$

where $c$ is the constant of integration

- These are given in the formula booklet
For the linear function $(a x + b)$ , where $a$ and $b$ are constants,

$\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + c$

$\int \frac{1}{a x + b} d x = \frac{1}{a} \ln | a x + b | + c$

It follows from the last result that

$\int \frac{a}{a x + b} d x = \ln | a x + b | + c$

- which can be deduced using Reverse Chain Rule
With ln, it can be useful to write the constant of integration, $c$ , as a logarithm
- using the laws of logarithms, the answer can be written as a single term
- $\int \frac{1}{x} d x = \ln | x | + \ln k = \ln k | x |$ where $k$ is a constant
- This is similar to the special case of differentiating $\ln (a x + b)$ when $b = 0$

Exam 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

Worked example

A curve has the gradient function $f' (x) = \frac{3}{3 x + 2} + e^{4 - x}$ .

Given the exact value of $f (1)$ is $\ln 10 - e^{3}$ find an expression for $f (x)$ .

NA5HYQ75_5-4-1-ib-sl-aa-only-we2-soltn

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Integrating Special Functions (DP IB Maths: AA HL)

Revision Note

How do I integrate sin, cos and sec^2?

How do I integrate exponentials and 1/x?

Author: Paul