Integrate from [0, 1/2]:
1/(4x^2+1)^(3/2) dx
To integrate the given function, we need to use a suitable substitution. Let's consider u = 4x^2 + 1.
Step 1: Find du/dx
Differentiating both sides of the equation u = 4x^2 + 1 with respect to x, we get:
du/dx = 8x
Step 2: Solve for dx
Rearranging the terms, we find:
dx = du / (8x)
Step 3: Substitute u and dx in the integral
The integral becomes:
∫ (1 / ((4x^2 + 1)^(3/2))) dx
= ∫ (1 / (u^(3/2))) * (du / (8x))
Step 4: Simplify the expression
Now, notice that we can simplify our expression by rearranging and factoring terms:
= (1 / 8) ∫ ((1 / (u^(3/2))) * (1 / x)) du
Step 5: Integrate using the substitution
To continue with the integration, we need to express everything in terms of u. Since u = 4x^2 + 1, we can replace x in the denominator with (1/2) * (√(u - 1)).
= (1 / 8) ∫ ((1 / (u^(3/2))) * (1 / (1/2) * √(u - 1))) du
= (1 / 8) ∫ ((2 / u^(3/2)) * (√(u - 1))) du
Step 6: Simplify further
We can simplify the expression by multiplying the constants and writing the exponents using negative powers:
= (1 / 4) ∫ (2u^(-3/2) * (u - 1)^(1/2)) du
Step 7: Evaluate the integral
Now, we can integrate term by term:
= (1 / 4) ∫ (2u^(-3/2) * (u - 1)^(1/2)) du
= (1 / 4) * (2/(-1/2)) * (u^(-1/2) * (u - 1)^(3/2)) + C
Step 8: Simplify the expression
= -1/2 * u^(-1/2) * (u - 1)^(3/2) + C
Step 9: Substitute back the value of u
Remember, u = 4x^2 + 1:
= -1/2 * (4x^2 + 1)^(-1/2) * ((4x^2 + 1) - 1)^(3/2) + C
= -1/2 * (4x^2 + 1)^(-1/2) * (4x^2)^(3/2) + C
= -1/2 * (4x^2 + 1)^(-1/2) * (8x^3) + C
So, the result of the definite integral from 0 to 1/2 is:
= [-1/2 * (4(1/2)^2 + 1)^(-1/2) * (8(1/2)^3)] - [-1/2 * (4(0)^2 + 1)^(-1/2) * (8(0)^3)]
= [-1/2 * (1/2)^(-1/2) * (8(1/2)^3)] - [-1/2 * (4(0)^2 + 1)^(-1/2) * (8(0)^3)]
= [-1/2 * (1/2)^(-1/2) * (8(1/2)^3)] - [-1/2 * (4(0)^2 + 1)^(-1/2) * (8(0)^3)]
= [-1/2 * √2 * (1/2)^(-3/2)] - [-1/2 * (1)^(-1/2) * (8(0)^3)]
= [-1/2 * √2 * 2^(-3/2)] - [-1/2 * 1 * 0]
= -√2/8.
Therefore, the value of the definite integral from 0 to 1/2 is -√2/8.