I posted the wrong problem an hour ago. What is the integration of √(cotx)/sinx dx by using substitution? I don't think it's solvable.
I used wolframalpha.c o m
and plugged
integrate √(cotx)/sinx dx
And it seems to integrate
sorry, don't know how to.
To solve the integral √(cotx)/sinx dx, you can use a substitution. Here's how:
Let's start by choosing u = cot(x). This will help us simplify the integral by transforming the variable of integration from x to u. To find du, we'll differentiate both sides of u = cot(x) with respect to x.
du/dx = -csc^2(x)
Now, we need to express dx in terms of du. We can rearrange the equation above to solve for dx:
dx = -du/csc^2(x)
Next, we'll substitute these expressions for u and dx into the original integral:
∫ √(cot(x))/sin(x) dx = ∫ √(u)/(-du/csc^2(x))
Simplifying further,
= ∫ -√(u) csc^2(x) du
At this point, you may think that the integral is unsolvable because it still contains x terms. However, since we have expressed dx in terms of du, we can also express csc^2(x) in terms of u.
Recall that csc(x) = 1/sin(x). So, csc^2(x) = (1/sin(x))^2 = 1/sin^2(x). We can further manipulate this as follows:
csc^2(x) = 1/sin^2(x) = 1/(1 - cos^2(x)) = sec^2(x)/(sec^2(x) - 1) = sec^2(x)/(tan^2(x))
Now, substituting this expression back into the integral:
∫ -√(u) csc^2(x) du = ∫ -√(u) (sec^2(x)/(tan^2(x))) du
= -∫ √(u) sec^2(x)/tan^2(x) du
Now, we need to find an expression for sec^2(x) and tan^2(x) in terms of u. Since u = cot(x) = 1/tan(x), we can rewrite these trigonometric functions as follows:
sec^2(x) = 1 + tan^2(x) = 1 + 1/u^2 = (u^2 + 1)/u^2
tan^2(x) = 1 - sec^2(x) = 1 - (u^2 + 1)/u^2 = (u^2 - 1)/u^2
Substituting these expressions back into the integral:
= -∫ √(u) [(u^2 + 1)/u^2] [(u^2 - 1)/u^2] du
= -∫ √(u) (u^2 + 1)(u^2 - 1)/u^4 du
= -∫ (u^2 + 1)(u^2 - 1)/u^(7/2) du
The integral at this stage may not appear easily solvable. However, it's important to note that this technique, known as substitution, does not always lead to an immediate "closed-form" solution.
In some cases, certain integrals can be evaluated using special functions or by applying additional techniques. At times, numerical methods such as approximations or computer algorithms are required to find a numerical solution.
Given the expression above, I can assist you with numerically evaluating the integral if you provide specific bounds or any additional requirements.