Answer:
![\displaystyle D) {x}^{5} + \rm C](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20D%29%20%20%7Bx%7D%5E%7B5%7D%20%20%2B%20%20%5Crm%20C)
Step-by-step explanation:
we would like to integrate the following Integral:
![\displaystyle \int 5 {x}^{4} \, dx](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%5Cint%205%20%7Bx%7D%5E%7B4%7D%20%5C%2C%20dx%20)
well, to get the constant we can consider the following Integration rule:
![\displaystyle \int c{x} ^{n} \, dx = c\int {x}^{n} \, dx](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%5Cint%20c%7Bx%7D%20%5E%7Bn%7D%20%20%5C%2C%20dx%20%20%3D%20%20c%5Cint%20%20%7Bx%7D%5E%7Bn%7D%20%20%5C%2C%20dx)
therefore,
![\displaystyle 5\int {x}^{4} \, dx](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%205%5Cint%20%20%7Bx%7D%5E%7B4%7D%20%5C%2C%20dx%20)
recall exponent integration rule:
![\displaystyle \int {x} ^{n} \, dx = \frac{ {x}^{n + 1} }{n + 1}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%5Cint%20%7Bx%7D%20%5E%7Bn%7D%20%20%5C%2C%20dx%20%20%3D%20%20%5Cfrac%7B%20%7Bx%7D%5E%7Bn%20%2B%201%7D%20%7D%7Bn%20%2B%201%7D%20)
so let,
Thus integrate:
![\displaystyle = 5\left( \frac{ {x}^{4+ 1} }{4 + 1} \right)](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%3D%20%205%5Cleft%28%20%5Cfrac%7B%20%7Bx%7D%5E%7B4%2B%201%7D%20%7D%7B4%20%2B%20%201%7D%20%20%5Cright%29)
simplify addition:
![\displaystyle = 5\left( \frac{ {x}^{5} }{5} \right)](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%3D%20%205%5Cleft%28%20%5Cfrac%7B%20%7Bx%7D%5E%7B5%7D%20%7D%7B5%7D%20%20%5Cright%29)
reduce fraction:
![\displaystyle = {x}^{5}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%3D%20%20%7Bx%7D%5E%7B5%7D%20)
finally we of course have to add the constant of integration:
![\displaystyle \boxed{ {x}^{5} + \rm C}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%20%5Cboxed%7B%20%7Bx%7D%5E%7B5%7D%20%20%2B%20%20%5Crm%20C%7D)
hence,
our answer is D)
Answer:
idk
Step-by-step explanation:
Answer:
1/8
Step-by-step explanation:
sin²(π/8) − cos⁴(3π/8)
Use power reduction formulas:
1/2 (1 − cos(2×π/8)) − 1/8 (3 + 4 cos(2×3π/8) + cos(4×3π/8))
Simplify:
1/2 (1 − cos(π/4)) − 1/8 (3 + 4 cos(3π/4) + cos(3π/2))
1/2 (1 − √2/2) − 1/8 (3 + 4 (-√2/2) + 0)
1/2 − √2/4 − 1/8 (3 − 2√2)
1/2 − √2/4 − 3/8 +√2/4
1/2 − 3/8
1/8