Answer:
∫
= ![\frac{1}{2}(x^6-2)^2+C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%28x%5E6-2%29%5E2%2BC)
Step-by-step explanation:
To find:
∫![6x^5(x^6-2)\,dx](https://tex.z-dn.net/?f=6x%5E5%28x%5E6-2%29%5C%2Cdx)
Solution:
Method of substitution:
Let ![x^6-2=t](https://tex.z-dn.net/?f=x%5E6-2%3Dt)
Differentiate both sides with respect to ![t](https://tex.z-dn.net/?f=t)
![6x^5\,dx=dt](https://tex.z-dn.net/?f=6x%5E5%5C%2Cdx%3Ddt)
[use
]
So,
∫
= ∫
=
where
is a variable.
(Use ∫
)
Put ![t=x^6-2](https://tex.z-dn.net/?f=t%3Dx%5E6-2)
∫
= ![\frac{1}{2}(x^6-2)^2+C_1](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%28x%5E6-2%29%5E2%2BC_1)
Use ![(a-b)^2=a^2+b^2-2ab](https://tex.z-dn.net/?f=%28a-b%29%5E2%3Da%5E2%2Bb%5E2-2ab)
So,
∫
= ![\frac{1}{2}(x^6-2)^2+C_1=\frac{1}{2}(x^{12}+4-4x^6)+C_1=\frac{x^{12} }{2}-2x^6+2+C_1=\frac{x^{12} }{2}-2x^6+C](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%28x%5E6-2%29%5E2%2BC_1%3D%5Cfrac%7B1%7D%7B2%7D%28x%5E%7B12%7D%2B4-4x%5E6%29%2BC_1%3D%5Cfrac%7Bx%5E%7B12%7D%20%7D%7B2%7D-2x%5E6%2B2%2BC_1%3D%5Cfrac%7Bx%5E%7B12%7D%20%7D%7B2%7D-2x%5E6%2BC)
where ![C=2+C_1](https://tex.z-dn.net/?f=C%3D2%2BC_1)
Without using substitution:
∫
= ∫
= ![\frac{6x^{12} }{12}-\frac{12x^6}{6}+C=\frac{x^{12} }{2}-2x^6+C](https://tex.z-dn.net/?f=%5Cfrac%7B6x%5E%7B12%7D%20%7D%7B12%7D-%5Cfrac%7B12x%5E6%7D%7B6%7D%2BC%3D%5Cfrac%7Bx%5E%7B12%7D%20%7D%7B2%7D-2x%5E6%2BC)
So, same answer is obtained in both the cases.
Answer:
72
Step-by-step explanation:
You use Pythagorean Theorem to to find the missing side which in this case is 6 and since and then from there you just add
Yes, distance can be a negative value
Answer:
inches
Step-by-step explanation:
I suppose they are asking you for the length of the red part.
18
*
=
inches
Answer:
Graph B will show his distance from the top of the mountain.
On Graph B, at 0 hours, the height of the graph would be 500.
Then, the graph would drop until 1 hour, when Enrique reaches the top. At this point, the height of Graph B would be 0.
Graph B would then rise as Enrique travels back down the mountain until the graph is at 500.
The words drop and rise can be interchanged for a synonym.