the integral from 0 to 1/4 of arcsin(x^(1/2))dx.
let z = x^.5
then dz = .5 x^-.5 dx
so dx = .5 x^.5 dz = .5 z dz
so
.5 z arcsin(z) dz
.5 z^2/2 sin^-1(z) -(1/4)sin^-1(z) +(z/4)sqrt(1-z^2)
or
.5 x sin^-1(x^.5) - (1/4)sin^-1(x^.5) + (1/4)x^.5 sqrt(1-x)
zero when x = 0
put in x = 1/4
dz = .5 x^-.5 dx
so dx = 2 x^.5 dz = 2 z dz
so
2 z arcsin(z) dz
etc
To evaluate the integral of arcsin(x^(1/2))dx from 0 to 1/4, we can use integration by substitution. Here's how you can proceed step-by-step:
Step 1: Identify the substitution variable.
Let u = x^(1/2). By substituting u into the integral, the expression becomes:
∫ arcsin(u) * (2u) du
Step 2: Differentiate and solve for dx.
Differentiating both sides of the equation u = x^(1/2), we get:
du = (1/2)x^(-1/2) dx
Rearranging the equation, we can solve for dx:
dx = 2u du
Step 3: Rewrite the integral.
Substitute the expression for dx from Step 2 into the integral, and also change the limits of integration:
∫ arcsin(u) * (2u) du
Now the limits of integration become u = √0 = 0 and u = √(1/4) = 1/2.
Step 4: Evaluate the integral.
Substitute the new limits of integration into the integral and solve it:
∫ [arcsin(u) * (2u) du] from 0 to 1/2
Step 5: Evaluate the integral using anti-differentiation.
To evaluate the integral of arcsin(u) * (2u), we can use integration by parts. Let's use u-substitution again with v = arcsin(u) and du = 2u du. Then:
∫ u dv = u*v - ∫ v du
By substituting back u and v, we get:
= u * arcsin(u) - ∫ arcsin(u) du
Step 6: Integrate u dv and ∫ arcsin(u) du.
∫ arcsin(u) du = -u * sqrt(1 - u^2) + ∫ sqrt(1 - u^2) du
The second integral on the right side is the integral of √(1 - u^2), which is the formula for evaluating the integral of a sine function.
Step 7: Calculate the integral of √(1 - u^2).
The integral of √(1 - u^2) is evaluated using the substitution u = sin(theta). The resulting integral is:
∫ sqrt(1 - u^2) du = ∫ cos(theta) d(theta) = sin(theta) + C
Substituting back u for sin(theta), we have:
sin(theta) + C = sin(arcsin(u)) + C = u + C
Step 8: Substitute back into the main integral and evaluate the result.
Now, substitute the value of the second integral back into the main integral:
u * arcsin(u) - ∫ arcsin(u) du = u * arcsin(u) - (-u * sqrt(1 - u^2) + ∫ sqrt(1 - u^2) du)
Simplifying further, we get:
u * arcsin(u) + u * sqrt(1 - u^2) - (u + C)
Finally, substitute back x^(1/2) for u:
= x^(1/2) * arcsin(x^(1/2)) + x^(1/2) * sqrt(1 - x^(1/2)^2) - (x^(1/2) + C)
This is the final result of the integral from 0 to 1/4 of arcsin(x^(1/2))dx.