·
Let's learn about the rules to intergration in A-Level Maths!
Power Rule
Example: What is ∫x3 dx ?
The question is asking "what is the integral of x3 ?"
We can use the Power Rule, where n=3:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/372048_731535.jpeg)
Multiplication by constant
Example: What is ∫6x2 dx ?
We can move the 6 outside the integral:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/587767_215897.png)
Trigonometry (x in radians)
Example: what is the integral of sin(x) ?
From the table above it is listed as being −cos(x) + C
It is written as:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/714416_106728.png)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/460874_784419.jpeg)
Sum Rule
Example:
What is ∫(cos x + x) dx ?
Use the Sum Rule:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/471073_586242.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/655903_958198.jpeg)
Reciprocal rule
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/524116_242532.jpeg)
Integration by Parts
∫u v dx = u∫v dx −∫u' (∫v dx) dx
- u is the function u(x)
- v is the function v(x)
- u' is the derivative of the function u(x)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/322806_38173.png)
Exponential rule
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/790428_616323.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/171359_788905.png)
Integration by Substitution
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/508692_365536.jpeg)
Note that we have g(x) and its derivative g'(x)
Example:
∫cos(x2) 2x dx
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/140118_521250.png)
dx2/dx = 2x which cancels out the 2x here.
So ∫cos(x2) 2x dx = sin(x2) + C
Drafted by Eunice (Maths)
References:
https://www.mathsisfun.com/calculus/integration-by-parts.html
https://www.mathsisfun.com/calculus/integration-rules.html