`int1/(xsqrt((1+x^n)))dx=`

Options

A. `log""(sqrt((1+x^n))-1)/(sqrt((1+x^n))+1)`
B. `log""(sqrt((1+x^n))+1)/(sqrt((1+x^n))-1)`
C. `1/nlog""(sqrt((1+x^n))-1)/(sqrt((1+x^n))+1)`
D. none

Answer

To solve the integral

\[ \int \frac{1}{x \sqrt{1 + x^n}} \, dx, \]

we can use substitution. Let's go through the steps:

### Step 1: Substitution
Let

\[ t = \sqrt{1 + x^n}. \]

Then, squaring both sides gives:

\[ t^2 = 1 + x^n \implies x^n = t^2 - 1. \]

### Step 2: Differentiate
Now we differentiate \(t\) with respect to \(x\):

\[ \frac{dt}{dx} = \frac{1}{2\sqrt{1 + x^n}} \cdot (n x^{n-1}) = \frac{n x^{n-1}}{2\sqrt{1 + x^n}}. \]

Rearranging gives:

\[ dx = \frac{2\sqrt{1 + x^n}}{n x^{n-1}} dt. \]

### Step 3: Substitute in the Integral
Now we substitute \(dx\) and \(t\) into the integral:

\[ \int \frac{1}{x \sqrt{1 + x^n}} \, dx = \int \frac{1}{x t} \cdot \frac{2\sqrt{1 + x^n}}{n x^{n-1}} dt. \]

Since \(\sqrt{1 + x^n} = t\), we can replace it:

\[ = \int \frac{1}{x t} \cdot \frac{2t}{n x^{n-1}} dt = \int \frac{2}{n x^n} dt. \]

### Step 4: Express \(x\) in terms of \(t\)
From \(x^n = t^2 - 1\), we have:

\[ x = (t^2 - 1)^{1/n}. \]

Thus,

\[ x^n = t^2 - 1 \implies x^n = t^2 - 1. \]

### Step 5: Substitute back into the Integral
Now we can rewrite the integral:

\[ \int \frac{2}{n(t^2 - 1)} dt. \]

### Step 6: Integrate
The integral

\[ \int \frac{2}{n(t^2 - 1)} dt \]

can be solved using the formula for the integral of the form \(\int \frac{1}{a^2 - x^2} dx\):

\[ \int \frac{1}{a^2 - x^2} dx = \frac{1}{2a} \ln \left| \frac{a+x}{a-x} \right| + C. \]

In our case, we have:

\[ = \frac{2}{n} \cdot \frac{1}{2} \ln \left| \frac{t + 1}{t - 1} \right| + C = \frac{1}{n} \ln \left| \frac{t + 1}{t - 1} \right| + C. \]

### Step 7: Substitute \(t\) back
Now we substitute back \(t = \sqrt{1 + x^n}\):

\[ = \frac{1}{n} \ln \left| \frac{\sqrt{1 + x^n} + 1}{\sqrt{1 + x^n} - 1} \right| + C. \]

### Final Answer
Thus, the final result of the integral is:

\[ \int \frac{1}{x \sqrt{1 + x^n}} \, dx = \frac{1}{n} \ln \left| \frac{\sqrt{1 + x^n} + 1}{\sqrt{1 + x^n} - 1} \right| + C. \]