Hello, I just wanted to verify if my work was good.
Calculate the following integral by parts:
∫ upper limit is 1/5 and lower limit is 1/10. of 10sin^-1 (5x)dx
so first I named the variables:
u = 10 sin^-1 (5x)
du = 50 / sqr(1-25x^2)
dv = dx
v = x
so we get:
= 10 sin^-1 (5x)(x) - ∫50x/(1-25x^2)
= 10 sin^-1 (5x)(x)|1/5, 1/10 -
∫50x/(1-25x^2) |1/5, 1/10
let w = 1-25x^2
dw = -50xdx
= 10 sin^-1 (5x)(x) + ∫ 1/sqr(w)dw
= 10 sin^-1 (5x)(x) + 2sqr(w) + C |1/5, 1/10
= 180 - (30 + 2sqr(0.75))
= 148.27
Thanks!
Your work seems mostly correct! However, there are a couple of mistakes in your calculations. Let's go through the steps together to find the correct solution:
To integrate by parts, we usually choose a function to differentiate (u) and another function to integrate (dv). In this case, you correctly chose:
u = 10 sin^⁻¹(5x)
dv = dx
Now, let's find du and v:
To find du, we need to differentiate u with respect to x:
du/dx = 10 / √(1 - (5x)^2)
Using the chain rule, we can rewrite this as:
du/dx = 10 / √(1 - 25x^2)
To find v, we integrate dv with respect to x:
v = ∫dx = x
Now, we can apply the integration by parts formula:
∫u dv = uv - ∫v du
Using the above formula, we have:
∫(10 sin^⁻¹(5x)) dx = (10 sin^⁻¹(5x))(x) - ∫ x [10 / √(1 - 25x^2)] dx
The remaining integral requires a u-substitution:
Let's substitute u = 1 - 25x^2, then du = -50x dx:
∫ x [10 / √(1 - 25x^2)] dx = -∫ (1/5) [-10 / √u] du
Now, we can simplify the integral:
-∫ (1/5) [-10 / √u] du = 2√u + C = 2 √(1 - 25x^2) + C
Now, let's evaluate the definite integral from the lower limit (1/10) to the upper limit (1/5):
= (10 sin^⁻¹(1))(1/5) - (10 sin^⁻¹(1/2))(1/10) + (2 √(1 - 25(1/5)^2)) - (2 √(1 - 25(1/10)^2))
= 2 + 2 + 0 - 0
= 4
So, the correct value for the integral is 4.
Please double-check the steps and try to identify the mistakes in your calculations.