`intsqrt(x^6+1).(log(x^6+1)-6logx)/(x^10)dx "is" =-(1)/(6)[(2)/(3)t^(3//2)logt-(4)/(3)t^(3//2)]+C` where t=

Options

A. `t=(1)/(x^6)`
B. `t=1+(1)/(x^6)`
C. `t=1-(1)/(x^6)`
D. None of these

Answer

To solve the integral

\[ \int \frac{\sqrt{x^6 + 1} \left( \log(x^6 + 1) - 6 \log x \right)}{x^{10}} \, dx, \]

we will follow these steps:

### Step 1: Simplify the Logarithmic Expression

Using the properties of logarithms, we can rewrite the expression inside the integral. Specifically, we can express \(6 \log x\) as \(\log(x^6)\) to combine the logs:

\[ \log(x^6 + 1) - 6 \log x = \log\left(\frac{x^6 + 1}{x^6}\right). \]

Thus, we can rewrite the integral as:

\[ \int \frac{\sqrt{x^6 + 1} \cdot \log\left(\frac{x^6 + 1}{x^6}\right)}{x^{10}} \, dx. \]

### Step 2: Rewrite the Integral

Now, we can express the integral in a more manageable form:

\[ \int \frac{\sqrt{x^6 + 1}}{x^{10}} \cdot \log\left(1 + \frac{1}{x^6}\right) \, dx. \]

### Step 3: Substitution

Let us make the substitution:

\[ t = 1 + \frac{1}{x^6}. \]

Then, we differentiate both sides:

\[ dt = -\frac{6}{x^7} \, dx \quad \Rightarrow \quad dx = -\frac{x^7}{6} \, dt. \]

From our substitution, we can express \(x^6\) in terms of \(t\):

\[ x^6 = \frac{1}{t - 1} \quad \Rightarrow \quad x = \left(\frac{1}{t - 1}\right)^{1/6}. \]

### Step 4: Change of Variables in the Integral

Now, substituting \(dx\) and \(x\) into the integral gives:

\[ \int \sqrt{\frac{1}{t - 1} + 1} \cdot \log(t) \cdot \left(-\frac{\left(\frac{1}{t - 1}\right)^{7/6}}{6}\right) \, dt. \]

This simplifies to:

\[ -\frac{1}{6} \int \sqrt{t} \cdot \log(t) \cdot dt. \]

### Step 5: Integration by Parts

Now we apply integration by parts. Let:

- \(u = \log(t)\) and \(dv = \sqrt{t} \, dt\).

Then we have:

- \(du = \frac{1}{t} \, dt\) and \(v = \frac{2}{3} t^{3/2}\).

Using integration by parts:

\[ \int u \, dv = uv - \int v \, du, \]

we get:

\[ -\frac{1}{6} \left( \log(t) \cdot \frac{2}{3} t^{3/2} - \int \frac{2}{3} t^{3/2} \cdot \frac{1}{t} \, dt \right). \]

### Step 6: Simplifying the Integral

The integral simplifies to:

\[ -\frac{1}{6} \left( \log(t) \cdot \frac{2}{3} t^{3/2} - \frac{2}{3} \int t^{1/2} \, dt \right). \]

Integrating \(t^{1/2}\):

\[ \int t^{1/2} \, dt = \frac{2}{3} t^{3/2}. \]

Putting it all together, we have:

\[ -\frac{1}{6} \left( \log(t) \cdot \frac{2}{3} t^{3/2} - \frac{2}{3} \cdot \frac{2}{3} t^{3/2} \right) + C. \]

### Final Step: Substitute Back

Finally, substituting back \(t = 1 + \frac{1}{x^6}\):

\[ = -\frac{1}{6} \left( \frac{2}{3} (1 + \frac{1}{x^6})^{3/2} \log(1 + \frac{1}{x^6}) - \frac{4}{9} (1 + \frac{1}{x^6})^{3/2} \right) + C. \]

### Conclusion

Thus, we find that

\[ t = 1 + \frac{1}{x^6}. \]