# 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.

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.

$$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx$$

Let $t=\sqrt{x} \Rightarrow \frac{dt}{dx}=\frac{1}{2\sqrt{x}} \Rightarrow 2 dt=\frac{dx}{\sqrt{x}}$

Can you take it from here?

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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?

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.

$$\int e^{\sqrt{x}}\frac{dx}{\sqrt x}$$

$$u=\sqrt x$$
$$du=\frac{dx}{2\sqrt x} \rightarrow 2du=\frac{dx}{\sqrt x}$$

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

$$2\int e^u du$$

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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.