`int(e^(2x)dx)/(root4(e^(x)-1))` equals

Options

A. `(4)/(21)(e^(x)-1)^(3//4)(3e^(x)+4)+c`
B. `((e^(x)-1)^(1//4)(3e^(x)+4))/(21)+c`
C. `(4)/(21)(e^(x)+1)(3e^(x)+4)`
D. None of these

Answer

To solve the integral

\[ I = \int \frac{e^{2x}}{\sqrt[4]{e^x - 1}} \, dx, \]

we can start by making a substitution to simplify the expression.

### Step 1: Substitution
Let \( e^x = t \). Then, \( dx = \frac{dt}{t} \).

### Step 2: Rewrite the Integral
Substituting \( e^x \) with \( t \), we have:

\[ e^{2x} = (e^x)^2 = t^2, \]

and the integral becomes:

\[ I = \int \frac{t^2}{\sqrt[4]{t - 1}} \cdot \frac{dt}{t} = \int \frac{t}{\sqrt[4]{t - 1}} \, dt. \]

### Step 3: Rewrite the Integral
Now, we can rewrite the integral as:

\[ I = \int t (t - 1)^{-1/4} \, dt. \]

### Step 4: Integration by Parts
We will use integration by parts. Let's set:

- \( u = t \) (thus \( du = dt \))
- \( dv = (t - 1)^{-1/4} dt \)

To find \( v \), we need to integrate \( dv \):

\[ v = \int (t - 1)^{-1/4} dt. \]

Using the power rule, we have:

\[ v = \frac{(t - 1)^{3/4}}{3/4} = \frac{4}{3} (t - 1)^{3/4}. \]

### Step 5: Apply Integration by Parts
Now we apply the integration by parts formula:

\[ I = uv - \int v \, du. \]

Substituting \( u \) and \( v \):

\[ I = t \cdot \frac{4}{3} (t - 1)^{3/4} - \int \frac{4}{3} (t - 1)^{3/4} dt. \]

### Step 6: Solve the Remaining Integral
Now we need to solve the integral \( \int (t - 1)^{3/4} dt \):

Using the power rule again:

\[ \int (t - 1)^{3/4} dt = \frac{(t - 1)^{7/4}}{7/4} = \frac{4}{7} (t - 1)^{7/4}. \]

### Step 7: Substitute Back
Now substituting back into our expression for \( I \):

\[ I = t \cdot \frac{4}{3} (t - 1)^{3/4} - \frac{4}{3} \cdot \frac{4}{7} (t - 1)^{7/4} + C. \]

### Step 8: Simplify
Combining terms, we have:

\[ I = \frac{4}{3} t (t - 1)^{3/4} - \frac{16}{21} (t - 1)^{7/4} + C. \]

### Step 9: Substitute Back for \( t \)
Finally, substituting back \( t = e^x \):

\[ I = \frac{4}{3} e^x (e^x - 1)^{3/4} - \frac{16}{21} (e^x - 1)^{7/4} + C. \]

### Final Answer
Thus, the integral evaluates to:

\[ I = \frac{4}{3} e^x (e^x - 1)^{3/4} - \frac{16}{21} (e^x - 1)^{7/4} + C. \]