Understanding IITJEE Integration: The Easy Way  1
Suppose you were told that a function had a gradient given by  
Could you work out that function? 
By referring back to differentiation, you would notice that this gradient was given by the function

Not necessarily! We would get the same gradient for the function



This is only the most basic use of integration. Together with differentiation, it is probably the most important area of maths.
The basic method of integration, is to reverse differentiation.
To differentiate, we multiplied by the power, then subtracted 1 from the power 


To integrate, we reverse this exactly.
First add 1 to the power, then divide by the new power. 

Notice that there is a constant added onto the function. This is because, as in the case above, we do not know whether or not a number must be added to our function.
Example
Find the curve whose gradient is 

Add 1 the power an divide by new power.  
Simplify  Important If you leave off the c, your answer is wrong and you will lose a mark 
If we are told a point that the graph passes through, then we can evaluate the constant.
For example
A curve has a gradient function  
and passes through the point 
Find the equation of the curve
Add 1 to the power and divide by new power.  
Simplify  


So the curve is 
What is the effect of integrating a constant?  We can say this because anything to the power 0 is equal to 1  
The power of x now becomes 1  


Which gives  When you integrate a constant, you just put an x behind the constant.  
The problem case  
What happens if you integrate?  
The power goes to nought, so we get  Not allowed!  
We can however integrate x^{1}  
Remember from differentiation that if  
then  
Since integration is just the reverse of differentiation, this is our answer.




say x=1 when y=2 