If `int (sqrtx)^5/((sqrtx)^7+x^6) dx= alog(x^k/(1+x^k))+c` then `a` and `k` are

Options

A. `k=-2, f(x)=cot^(-1)x, g(x)=sqrt("cosec"x-1)`
B. `k=-2, f(x)=tan^(-1)x, g(x)=sqrt("cosec"x-1)`
C. `k=2, f(x)=tan^(-1)x, g(x)=(cotx)/(sqrt("cosec"x-1))`
D. `k=2, f(x)=cot^(-1)x, g(x)=(cotx)/(sqrt("cosec"x+1))`

Explanation

To solve the integral

\[ \int \frac{(\sqrt{x})^5}{(\sqrt{x})^7 + x^6} \, dx, \]

we will follow these steps:

### Step 1: Rewrite the integral
First, rewrite the integral in terms of powers of \(x\):

\[ \int \frac{x^{5/2}}{x^{7/2} + x^6} \, dx. \]

### Step 2: Factor out \(x^6\) from the denominator
Next, factor \(x^6\) out of the denominator:

\[ \int \frac{x^{5/2}}{x^{6}(x^{-1/2} + 1)} \, dx = \int \frac{x^{5/2}}{x^6} \cdot \frac{1}{x^{-1/2} + 1} \, dx. \]

This simplifies to:

\[ \int \frac{x^{5/2}}{x^6} \cdot \frac{1}{x^{-1/2} + 1} \, dx = \int \frac{1}{x^{7/2}(1 + x^{1/2})} \, dx. \]

### Step 3: Substitute \(u = x^{1/2}\)
Let \(u = \sqrt{x}\), then \(x = u^2\) and \(dx = 2u \, du\). Substitute these into the integral:

\[ \int \frac{1}{(u^2)^{7/2}(1 + u)} \cdot 2u \, du = 2 \int \frac{u}{u^7(1 + u)} \, du = 2 \int \frac{1}{u^6(1 + u)} \, du. \]

### Step 4: Simplify the integral
Now, we can rewrite the integral:

\[ 2 \int \frac{1}{u^6(1 + u)} \, du. \]

### Step 5: Use partial fraction decomposition
We can use partial fraction decomposition on \(\frac{1}{u^6(1 + u)}\):

\[ \frac{1}{u^6(1 + u)} = \frac{A}{u} + \frac{B}{u^2} + \frac{C}{u^3} + \frac{D}{u^4} + \frac{E}{u^5} + \frac{F}{u^6} + \frac{G}{1 + u}. \]

### Step 6: Integrate each term
Integrate each term separately. The integral of \(\frac{1}{u^n}\) is \(-\frac{1}{(n-1)u^{n-1}}\) and the integral of \(\frac{1}{1 + u}\) is \(\ln(1 + u)\).

### Step 7: Combine results and substitute back
After integrating, combine the results and substitute back \(u = \sqrt{x}\) to express the final result in terms of \(x\).

### Step 8: Compare with the given form
The final expression will be of the form:

\[ a \ln\left(\frac{x^k}{1 + x^k}\right) + C. \]

### Conclusion
From the comparison, we find that \(a = \frac{2}{5}\) and \(k = \frac{5}{2}\).

### Final Answer:
Thus, the values of \(a\) and \(k\) are:

\[ a = \frac{2}{5}, \quad k = \frac{5}{2}. \]