"Thinking backwards" shows up in a lot of places. That means it must be important! Here, "thinking backwards" means instead of using a known integral value to evaluate an expression, we'll work backwards from an equation to find the integral value.

Think of the integral you're looking for as a unit. Instead of solving for *x*, you'll be solving for something like *. *If your equation contains a slightly different integral than the one you want, use the properties to manipulate the integral until it's what you're looking for.

## Practice:

Find given that
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Pull out the constant from the integral to get We're trying to find , which means we need to get that all by itself on one side of the equation. Divide both sides by 2, and there you are:
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Find given that
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Split up the integral: We know , so
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Combine the tricks used in the previous two examples to find given that
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Split up the integral and pull out the constant 5 from the second term: We know so we can put that in to get Since *,* it's time to solve:
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Find assuming that f is even and that
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Since *f* is even, Since this needs to be 16, must be 8. | |

Find assuming that f is even and that
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Switch the limits, switch the signs: Since *f* is even, this equals So *which means .*
*Whenever you're dealing with an odd function, you want to be on the lookout for integrals of the form ,because you know that's equal to 0.*
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Find given that

Answer

We can combine the integrals to get

Then pull out the constant to get

Since this must equal 48,

Find given that

*Answer*

*Split up the integral and pull out the constants:*

*We can find by looking at a graph. The region between **x* and the *x*-axis on [0,3] is a triangle with base 3 and height 3, so its area is 4.5.

*Since this must equal 39,*

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Find given that *f* (*x*) is even and that

Answer

Split up the integral, then evaluate the integral of the constant (don't be fooled by the order of the limits):

Since we know this whole mess must equal 34,

Switch the limits and switch the signs, then use the fact that *f* is even:

We conclude that .

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*Find given that *f* is odd and

Answer

Split up the integral, pull out the constant, and evaluate the integral of 4:

Since this needs to equal 26, we'll clean some things up.

Now switch the limits and switch the sign:

Split up the integral:

Since *f* is odd, its integral from -2 to 2 is zero, so

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