Evaluating Integral: ∫ (e√x / √x) dx

In summary: CIn summary, when faced with the integral ∫ (e√x / √x) dx, the conversation discusses using u-substitution to simplify the problem. The solution involves substituting u = √x and using the derivative to transform the integral into a simpler form. The final answer is 2e^u + C.
  • #1
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If you are told to evaluate the integral and are given this problem:

∫ (e√x / √x) dx

This reads that the integral of (e) to the power of square root of x divided by the square root of x multiplied by the derivative of x is……

I have tried solving it and came up with this, is this correct if not please show me what I did wrong and how I can solve it.

y = x2 + 2

y = sin x

0 ≤ x ≤ π
π π π π
A(D) = ∫0 (x2 + 2 – sin x) dx = ∫0 x2dx + 2 ∫0 dx - ∫0 sin x dx =

π π π
[x 3 / 3 ]0 + 2[x]0 + [cos x]0 = π/3 + 2π – 2

π
[cos x]0 = cos π – cos 0 = -1 – 1 = -2

Is this correct?

the integral goes from zero to pi. and x2 is x squared. Any help is appreciated.
 
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  • #2
[tex]\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx[/tex]

Let [itex]t=\sqrt{x} \Rightarrow \frac{dt}{dx}=\frac{1}{2\sqrt{x}} \Rightarrow 2 dt=\frac{dx}{\sqrt{x}}[/itex]

Can you take it from here?
 
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  • #3
now do you turn the square root of x into the power of 1/2, then you raise it to the numerator it becomes -1/2, the direvative of that is 1/2 divided by one whic is one half. correct?
 
  • #4
No ... if you u-sub the sqrt of x in the numerator and take the derivative, a part of your integral will appear which let's you get rid of the one in the denominator.
 
  • #5
[tex]\int e^{\sqrt{x}}\frac{dx}{\sqrt x}[/tex]

[tex]u=\sqrt x[/tex]
[tex]du=\frac{dx}{2\sqrt x} \rightarrow 2du=\frac{dx}{\sqrt x}[/tex]

Now make your substitutions ... hope this is a little more clear.

[tex]2\int e^u du[/tex]
 
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1. What is the purpose of evaluating an integral?

Evaluating an integral allows us to find the area under a curve or the value of a function over a certain interval. This can be useful in many fields, such as physics, engineering, and economics.

2. What is the meaning of the notation ∫ (e√x / √x) dx?

The notation ∫ (e√x / √x) dx represents the integral of the function e√x / √x with respect to the variable x. This means we are finding the area under the curve of this function over a certain interval of x values.

3. How do you evaluate an integral like ∫ (e√x / √x) dx?

To evaluate this integral, we need to use integration techniques such as substitution, integration by parts, or partial fractions. These techniques help us to simplify the integral and find a solution.

4. Can the integral ∫ (e√x / √x) dx be evaluated using basic calculus rules?

No, this integral cannot be evaluated using basic calculus rules. It requires more advanced techniques like substitution or integration by parts.

5. What is the result of evaluating ∫ (e√x / √x) dx?

The result of evaluating ∫ (e√x / √x) dx will be an expression involving x, such as a polynomial or a logarithmic function. The exact solution will depend on the integration technique used and the limits of integration.

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