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.
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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
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Find the curve whose gradient is |
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| Add 1 the power an divide by new power. |  |
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| 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. |  |
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| 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? |  |
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| The power goes to nought, so we get |  |
Not allowed! | ||
| We can however integrate x-1 | ||||
| Remember from differentiation that if |  |
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| then |  |
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| Since integration is just the reverse of differentiation, this is our answer.
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| say x=1 when y=2 |  |