Evaluate integral of e^x^(1/2) / x^(1/2)
I've looked at the answer but I don't understand what people do in their steps.
When I substitute x^(1/2) for u, I get:
2du = 1/x^(1/2) dx
But what do you do with the 1/x^(1/2) dx? It just disappears in the solutions I've seen people give.
yes. the substitution is correct:
let u = x^(1/2)
thus du = 1/[2(x^(1/2))] dx, or
dx = 2(x^(1/2)) du, or
dx = 2u du
substituting these to original integral,
integral of [e^x^(1/2) / x^(1/2)] dx
integral of [(e^u) / u] * (2u) du
the u's will cancel out:
integral of [2*e^u] du
we can readily integrate this to
2*e^u + C
substituting back the value of u,
2*e^(x^(1/2)) + C
hope this helps~ :)
This helped alot :)
To evaluate the integral of e^x^(1/2) / x^(1/2), you can use a substitution. Let's go through the steps.
Step 1: Substitute u = x^(1/2). This means that x = u^2.
Step 2: Find the derivative of u with respect to x to obtain du/dx = (1/2)x^(-1/2).
Step 3: Rearrange this equation to solve for dx: dx = 2u du.
Now, let's substitute these values back into the original integral:
∫ e^x^(1/2) / x^(1/2) dx
∫ e^u (1/x^(1/2)) dx // Substituting x = u^2
∫ e^u (1/u) (1/x^(1/2)) (2u du) // Substituting dx = 2u du
Now, we simplify the expression:
∫ 2e^u du
Step 4: Integrate with respect to u:
∫ 2e^u du = 2 ∫ e^u du = 2e^u + C,
where C is the constant of integration.
Step 5: Replace u with x^(1/2):
2e^u + C = 2e^(x^(1/2)) + C,
which is the final result of the integral.
So, in the solutions you mentioned, the 1/x^(1/2) dx term did not disappear; it was just substituted and simplified using the substitution and the derived value of dx.