The value of `int(e^(sqrtx))/(sqrtx(1+e^(2sqrtx)))dx` is equal to (where, C is the constant of integration)

Options

A. `tan^(-1)(2e^(sqrtx))+C`
B. `ln((1+e^(x))/(1-e^(sqrtx)))+C`
C. `2tan^(-1)(e^(sqrtx))+C`
D. `(tan^(-1)x)e^(sqrtx)+C`

Answer

To solve the integral \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}(1 + e^{2\sqrt{x}})} \, dx \), we will follow a systematic approach.

### Step-by-step Solution:

1. **Substitution**: Let \( t = \sqrt{x} \). Then, \( x = t^2 \) and \( dx = 2t \, dt \).

**Hint**: Substituting a simpler variable can often simplify the integral.

2. **Rewrite the Integral**: Substitute \( t \) into the integral:
\[ \int \frac{e^{\sqrt{x}}}{\sqrt{x}(1 + e^{2\sqrt{x}})} \, dx = \int \frac{e^t}{t(1 + e^{2t})} \cdot 2t \, dt \]
This simplifies to:
\[ 2 \int \frac{e^t}{1 + e^{2t}} \, dt \]

**Hint**: Always simplify the integral after substitution to make it easier to integrate.

3. **Further Simplification**: Notice that \( 1 + e^{2t} = 1 + (e^t)^2 \). We can use the substitution \( u = e^t \), which gives \( du = e^t \, dt \) or \( dt = \frac{du}{u} \).

**Hint**: Changing variables can help to transform the integral into a more manageable form.

4. **Substituting in the Integral**:
\[ 2 \int \frac{u}{1 + u^2} \cdot \frac{du}{u} = 2 \int \frac{1}{1 + u^2} \, du \]

**Hint**: When substituting, ensure that all parts of the integral are accounted for correctly.

5. **Integrate**: The integral \( \int \frac{1}{1 + u^2} \, du \) is a standard integral that equals \( \tan^{-1}(u) + C \).

**Hint**: Familiarize yourself with standard integrals to speed up the integration process.

6. **Back Substitute**: Replace \( u \) back with \( e^t \) and then \( t \) back with \( \sqrt{x} \):
\[ 2 \tan^{-1}(e^{\sqrt{x}}) + C \]

**Hint**: Always remember to revert back to the original variable after integration.

### Final Answer:
Thus, the value of the integral is:
\[ \int \frac{e^{\sqrt{x}}}{\sqrt{x}(1 + e^{2\sqrt{x}})} \, dx = 2 \tan^{-1}(e^{\sqrt{x}}) + C \]