Integrating Special Functions (DP IB Maths: AI HL)

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Author

Paul

Last updated

19 July 2024

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Integrating Trig Functions

How do I integrate sin, cos and 1/cos²?

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 $\tan x$

$\int \frac{1}{\cos^{2} x} d x = \tan x + c$

All three of these standard integrals are in the formula booklet
For the linear function $a x + b$ , where $a$ and $b$ are constants,

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

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

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

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

Examiner 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

Find, in the form

F (x) + c

, an expression for each integral

$\int \cos x d x$
$\int \frac{1}{\cos^{2} (3 x - \frac{π}{3})} d x$

5-4-1-ib-hl-ai-aa-extraaa-ai-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 $b = 0$

Examiner Tip

When revising, familiarise yourself with the layout of this section of the formula booklet, make sure you know what is and isn't in there and how to find it very quickly

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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