`intsqrt(5+x^(10))/x^(16)dx` is equal to

Options

A. `-1/75(1+5/x^(10))+c`
B. `-1/50(1+5/x^(10))^(3/2)+c`
C. `-1/50(1+5/x^(10))^(1/2)+c`
D. `-1/75(1+5/x^(10))^(3/2)+c`

Answer

To solve the integral \(\int \frac{\sqrt{5 + x^{10}}}{x^{16}} \, dx\), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\sqrt{5 + x^{10}}}{x^{16}} \, dx \]

### Step 2: Factor out \(x^{10}\)
We can factor \(x^{10}\) out of the square root:
\[ I = \int \frac{\sqrt{x^{10}(1 + \frac{5}{x^{10}})}}{x^{16}} \, dx \]
This simplifies to:
\[ I = \int \frac{x^{5}\sqrt{1 + \frac{5}{x^{10}}}}{x^{16}} \, dx = \int \frac{\sqrt{1 + \frac{5}{x^{10}}}}{x^{11}} \, dx \]

### Step 3: Substitute
Let \(u = 1 + \frac{5}{x^{10}}\). Then, we differentiate \(u\) with respect to \(x\):
\[ du = -\frac{50}{x^{11}} \, dx \quad \Rightarrow \quad dx = -\frac{x^{11}}{50} \, du \]

### Step 4: Substitute in the Integral
Now we substitute \(u\) and \(dx\) into the integral:
\[ I = \int \sqrt{u} \left(-\frac{x^{11}}{50}\right) \cdot \frac{1}{x^{11}} \, du = -\frac{1}{50} \int \sqrt{u} \, du \]

### Step 5: Integrate
Now we integrate \(\sqrt{u}\):
\[ -\frac{1}{50} \int u^{1/2} \, du = -\frac{1}{50} \cdot \frac{u^{3/2}}{3/2} = -\frac{2}{150} u^{3/2} = -\frac{1}{75} u^{3/2} \]

### Step 6: Substitute Back
Now we substitute back \(u = 1 + \frac{5}{x^{10}}\):
\[ I = -\frac{1}{75} \left(1 + \frac{5}{x^{10}}\right)^{3/2} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{\sqrt{5 + x^{10}}}{x^{16}} \, dx = -\frac{1}{75} \left(1 + \frac{5}{x^{10}}\right)^{3/2} + C \]