# Thread: Strange question involving definite integrals

1. ## Strange question involving definite integrals

Let .
Find
and

How can I go about solving this? I've done examples on definite integrals and did my note-reading but nowhere does it explain how to deal with this kind of problem. Any help is appreciated!

2. Originally Posted by Archduke01
Let .
Find
and

How can I go about solving this? I've done examples on definite integrals and did my note-reading but nowhere does it explain how to deal with this kind of problem. Any help is appreciated!
properties of definite integrals you should already be familiar with ...

$\int_a^b f(x) \pm g(x) \, dx = \int_a^b f(x) \, dx \pm \int_a^b g(x) \, dx$

$\int_a^b k \cdot f(x) \, dx = k \int_a^b f(x) \, dx$

$\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$

$\int_a^b f(x) \, dx + \int_b^c f(x) \, dx + \int_c^d f(x) \, dx = \int_a^d f(x) \, dx$

3. How do I use the properties to solve the problems?

4. use this property plus the given values for the various definite integrals of f(x) ...

$\int_a^b f(x) \, dx + \int_b^c f(x) \, dx + \int_c^d f(x) \, dx = \int_a^d f(x) \, dx$

... to find $\int_{5.5}^8 f(x) \, dx$

5. Thank you! I got the first one.

For the second, I notice that the upper limit is actually a smaller value than the lower. So that means I can flip them and make the integral negative? Even so, the 6f(x)-10 throws me off... Do I replace the f(x) with -10 (which was the answer to the previous question) and solve?

6. Originally Posted by Archduke01
Thank you! I got the first one.

For the second, I notice that the upper limit is actually a smaller value than the lower. So that means I can flip them and make the integral negative? Even so, the 6f(x)-10 throws me off... Do I replace the f(x) with -10 (which was the answer to the previous question) and solve?
properties, properties, properties ...

$\int_8^{5.5} 6f(x) - 10 \, dx = \int_8^{5.5} 6f(x) \, dx - \int_8^{5.5} 10 \, dx = 6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx
$

7. $6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx$

Thank you, but do I flip the a and b values now and make the integrals negative/positive? Because one of the properties state that.

8. Please respond. I don't know how to proceed.

9. $6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx = -6 \int_{5.5}^8 f(x) \, dx + \int_{5.5}^8 10 \, dx$

now finish it.

10. $6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx = -6
\int_{5.5}^8 f(x) \, dx + \int_{5.5}^8 10 \, dx
$

Since
$-6 \int_{5.5}^8 f(x)$ = -10, the equation becomes

$-6 * -10 + \int_{5.5}^8 10 \, dx$

What can I do to the second one to change that into a number too?

11. Wrong pleace, sorry.

12. I'm sorry, I don't understand. How was that formula relevant to my question?