Hello!
Sure, I can help you with that. The integral you've given is:
∫(0 to 1) 14x³√(1 - x²) dx
Let's perform the integration step by step:
1. Use substitution: Let u = 1 - x². Then, du/dx = -2x, and dx = du / (-2x).
2. Substitute the limits of integration as well: When x = 0, u = 1; when x = 1, u = 0.
Now the integral becomes:
∫(1 to 0) 14x³√u * (du / -2x)
Cancel out the x term:
∫(1 to 0) -7x²√u du
Integrate with respect to u:
[-7/3 * u^(3/2)] from 1 to 0
Substitute back u = 1 - x²:
-7/3 * (1 - x²)^(3/2) - (-7/3 * 1^(3/2))
Simplify:
-7/3 * (1 - x²)^(3/2) + 7/3
So, the value of the integral is:
-7/3 * (1 - x²)^(3/2) + 7/3
Hope this help you!
If you have more questions, feel free to ask!