Find the indefinite integral of (1-sqrtx)/(1+sqrtx)dx
I just worked completely through this problem using substitution (u=1+sqrtx) and came to the answer 3+4sqrtx+x+C.
This is incorrect. Not sure where I messed up.
well, first multiply top and bottom by
(1 - sqrt x)
(1-sqrt x)^2
------------ dx
1 - x
(1- 2 sqrt x )^2
------------ dx
1 - x
dx/(1-x) - 2 x^.5 dx/(1-x) + 4 x dx/(1-x)
now go to
http://www.wolframalpha.com/input/?i=integrate+x%2F%281-x%29
To find the indefinite integral of the given expression, we can use substitution as you mentioned. However, it seems there may have been a mistake in your calculations. Let's go through the problem step-by-step.
Let's start with the given integral:
∫[(1 - √x)/(1 + √x)] dx
To simplify the integral, let's introduce a substitution. Let u = √x, which implies u^2 = x. Taking the derivative of both sides with respect to x gives 2u du = dx.
Now, let's substitute these values back into the integral:
∫[(1 - u)/(1 + u)] * (2u du)
Simplifying further:
2∫[(u - 1)/(u + 1)] * u du
The integral can be expanded using partial fraction decomposition:
2∫(u - 1) du/(u + 1) + 2∫u du
Integrating each term separately:
2∫(u - 1) du = 2(u^2/2 - u) = u^2 - 2u
2∫u du = u^2
Substituting back the value of u = √x:
2(√x^2 - 2√x) + √x^2 + C
= 2x - 4√x + x + C
= 3x - 4√x + C
So, the correct indefinite integral of (1 - √x)/(1 + √x) dx is 3x - 4√x + C.
To find the indefinite integral of the given function (1 - sqrt(x))/(1 + sqrt(x)) dx, let's go through the steps together to see where the mistake may have occurred.
Start by letting u = sqrt(x) so that we can rewrite the integral in terms of u. By doing this, we can eliminate the square root and simplify the expression.
From the given substitution, we can rewrite x = u^2.
Next, differentiate both sides of the equation x = u^2 with respect to u to get dx = 2u du.
Now, substitute x and dx in terms of u into the integral:
∫ (1 - sqrt(x))/(1 + sqrt(x)) dx = ∫ (1 - u)/(1 + u) * 2u du.
Simplify the expression inside the integral:
= 2 ∫ (1 - u)/(1 + u) u du.
Expand the numerator:
= 2 ∫ (u - u^2)/(1 + u) du.
Divide the numerator by the denominator:
= 2 ∫ (u - u^2)/(1 + u) du = 2 ∫ (u - u^2)*(1 + u)^(-1) du.
Now, split the integral into two separate integrals:
= 2 ∫ u*(1 + u)^(-1) du - 2 ∫ u^2*(1 + u)^(-1) du.
Evaluate the first integral:
= 2 ∫ u/(1 + u) du.
For this integral, you can use the substitution method again. Let v = 1 + u, which gives dv = du. Rearranging the equation, we get u = v - 1.
Substituting the new variables into the integral, we have:
= 2 ∫ (v - 1)/v dv.
Simplify the integrand:
= 2 ∫ (1 - 1/v) dv = 2(v - ln|v|) + C, where C is the constant of integration.
Now evaluate the second integral:
= -2 ∫ u^2/(1 + u) du.
Using the substitution method once more, let w = 1 + u, which implies dw = du. Rearranging the equation, we get u = w - 1.
Substituting the new variables into the integral, we have:
= -2 ∫ (w - 1)^2/w dw.
Expand the numerator:
= -2 ∫ (w^2 - 2w + 1)/w dw.
Split the integral into three separate integrals:
= -2 ∫ w^2/w dw + 2 ∫ 2w/w dw - 2 ∫ 1/w dw.
Simplify the integrals:
= -2 ∫ w dw + 2 ∫ 2 dw - 2 ln|w| + C.
Evaluate the integrals:
= -w^2 + 2w - 2 ln|w| + C.
Substituting w = 1 + u:
= -(1 + u)^2 + 2(1 + u) - 2 ln|1 + u| + C.
Now, substitute the value of u back in:
= -(1 + sqrt(x))^2 + 2(1 + sqrt(x)) - 2 ln|1 + sqrt(x)| + C.
Simplifying this expression further may yield a different form, but it is equivalent to the correct indefinite integral.