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!
You're ok to this point:
10 sin^-1 (5x)(x) + 2√(1-25x^2) + C |1/5, 1/10
By this time you should realize that radians are the measure of choice for trig stuff.
sin^-1(1/2) = pi/6
sin^-1(1) = pi/2
so you end up with
[10(1/5 * pi/2) + 2√(1-1)] - [10(1/10 * pi/6) + 2√(1-1/4)]
pi - (pi/6 + √3)
5pi/6 - √3
I think you dropped a square root and a dx in
so we get:
= 10 sin^-1 (5x)(x) - ∫50x/(1-25x^2)
I got
= 10 sin^-1 (5x)(x) - ∫50x/(1-25x^2)^(1/2) dx
that last part can be integrated as
-2(1 - 25x^2)^(1/2)
or -2√(1-25x^2)
so your final integral answer would be
10x sin^-1 (5x) - 2(1-25x^2)^(1/2)
see if that works for you.
THanks Steve and Reiny!
To verify if your work is correct, let's go through the steps of evaluating the integral using integration by parts.
You correctly named the variables:
u = 10 sin^⁻¹(5x)
du = 50 / √(1 - 25x²)
dv = dx
v = x
Next, you applied the integration by parts formula:
∫ u dv = uv - ∫ v du
Applying this formula, we have:
∫ 10 sin^⁻¹(5x) dx = (10 sin^⁻¹(5x))(x) - ∫ x(50 / √(1 - 25x²)) dx
Simplifying the equation, we get:
= 10x sin^⁻¹(5x) - 50∫ (x² / √(1 - 25x²)) dx
Now, you introduced the substitution:
Let w = 1 - 25x²
dw = -50x dx
Using this substitution, the integral becomes:
-50∫ (x² / √(1 - 25x²)) dx = -∫ (x² / √w) dw
Now we can proceed to integrate:
= -2∫ w^(-1/2) dw
= -2(2√w) + C
= -4√w + C
Substituting back for w, we have:
= -4√(1 - 25x²) + C
Now, evaluating the definite integral with the given limits:
= [-4√(1 - 25x²)] from 1/10 to 1/5
Substituting the upper limit, 1/5:
= [-4√(1 - 25(1/5)²) ] - [-4√(1 - 25(1/10)²)]
= -4√(1 - 1/5²) + 4√(1 - 1/10²)
= -4√(1 - 1/25) + 4√(1 - 1/100)
Simplifying further:
= -4√(24/25) + 4√(99/100)
= -4(√24/√25) + 4(√99/√100)
= -4(√24/5) + 4(√99/10)
Finally, evaluating the expression:
= -0.8√24 + 0.4√99
Therefore, the correct value of the integral is approximately -0.8√24 + 0.4√99.
It seems like there was an error in your calculation. Please double-check your work to ensure accuracy.