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
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Not necessarily! We would get the same gradient for the function
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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 |
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| To integrate, we reverse this exactly. First add 1 to the power, then divide by the new power. |
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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 |
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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 |  | |||
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| 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 |  | |||
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| Which gives |  | When you integrate a constant, you just put an x behind the constant. | ||
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| 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. | ||||
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| say x=1 when y=2 |  |