IB Mathematics AA HL Flashcards SL 5.10 Indefinite integral of xn , sinx, cosx, and ex

Basic Integrals
On top of our first rule of \(\int x^n dx = \frac{x^{n+1}}{n+1} + c\), we can now see 4 rules for some other basic functions:

– \(\int \sin x dx = -\cos x + c\)
– \(\int \cos x dx = \sin x + c\)
– \(\int \sec^2 x dx = \tan x + c\)
– \(\int e^x dx = e^x + c\)

[q]

Example :
Find \(\int 4x^3 + \frac{1}{x} dx\):

– \(\int 4x^3 + \frac{1}{x} dx = x^4 + \ln |x| + c = x^4 + \ln |x| + c\)

[a]

“ax + b” Rules
If you remember what happens when differentiating with the chain rule with functions in the form \(f(ax+b)\), such as:

– \(\frac{d}{dx}(\sin(5x+2)) = 5\cos(5x+2)\) or \(\frac{d}{dx}((6x – 7)^3) = 3(6x – 7)^2 \times 6\), i.e. \(\frac{d}{dx} f(ax + b) = a f'(ax + b)\).

The opposite of multiplying by ‘a’ when differentiating is to divide by ‘a’ when integrating \(f(ax + b)\). If \(f(x) = F'(x) + c\), then \(\int f(ax + b) dx = \frac{1}{a} F(ax + b) + c\).

(Not in Formula Book)

[q]
Substitution
If \(\frac{d}{dx} F(g(x)) = f(g(x)) \cdot g'(x)\), then also: \(\int f(g(x)) \cdot g'(x) dx = F(g(x)) + c\).

So you can do a reverse chain-rule for integration, whenever you see a function \(g(x)\) and its derivative \(g'(x)\). What will make it easier is to substitute \(g(x) = u\), and considering \(\frac{du}{dx} = g'(x)\), simplifying, then doing \(\int f(u) du\).

[a]
Example :
Find \(\int x^3 \sec^2 (x^2) dx\):

– Let \(u = x^2 + 1\), then \(\frac{du}{dx} = 4x^3 dx\). Rearrange to \(\frac{1}{4} du = x^3 dx\), all this means that:
– \(\int (x^3 \cdot \sec^2 (x^2)) dx = \frac{1}{4} \int \sec^2 u du = \frac{1}{4} \tan u + c\).
– Substitute back: \( \frac{1}{4} \tan (x^2 + 1) + c\).

(Multiple Uses) Sometimes you fill in the \(\int f’g dx\) part, and that itself will need integration by parts to complete. Make sure you keep everything very organized.

[x] Exit text