If `int((sqrt(x)^(5))dx)/((sqrt(x))^(7)+x^(6))=lambdalog((x^(a))/(x^(a)+1))+C` then `a+lambda` equal to

Options

A. 2
B. `gt2`
C. `lt2`
D. `gt3`

Answer

To solve the given integral problem, we start with the expression:

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

### Step 1: Simplifying the Denominator
We can factor out \(x^6\) from the denominator:

\[ I = \int \frac{\sqrt{x}^5 \, dx}{x^6 \left(\frac{\sqrt{x}^7}{x^6} + 1\right)} = \int \frac{\sqrt{x}^5 \, dx}{x^6 \left(\frac{\sqrt{x}^7}{x^6} + 1\right)} = \int \frac{\sqrt{x}^5 \, dx}{x^6 \left(\sqrt{x}^{\frac{7}{2}} + 1\right)} \]

### Step 2: Rewriting the Integral
Now we rewrite the integral:

\[ I = \int \frac{x^{\frac{5}{2}} \, dx}{x^6 \left(x^{\frac{7}{12}} + 1\right)} = \int \frac{x^{\frac{5}{2}} \, dx}{x^6 \left(x^{\frac{7}{12}} + 1\right)} = \int \frac{dx}{x^{\frac{7}{2}} + x^6} \]

### Step 3: Substituting Variables
Let’s use the substitution \(t = \sqrt{x}\), which implies \(x = t^2\) and \(dx = 2t \, dt\):

\[ I = \int \frac{2t^5 \, dt}{t^7 + t^{12}} = 2 \int \frac{t^5 \, dt}{t^7 + t^{12}} \]

### Step 4: Simplifying Further
Now we can simplify the integral:

\[ I = 2 \int \frac{t^5 \, dt}{t^5(t^2 + t^7)} = 2 \int \frac{dt}{t^2 + t^7} \]

### Step 5: Integrating
The integral can be solved using partial fractions or a suitable substitution. However, we can also recognize that it resembles a logarithmic form:

\[ I = 2 \int \frac{dt}{t^2 + 1} \]

This integral evaluates to:

\[ I = 2 \tan^{-1}(t) + C \]

### Step 6: Back Substituting
Returning to the variable \(x\):

\[ I = 2 \tan^{-1}(\sqrt{x}) + C \]

### Step 7: Matching with Given Expression
We need to match this with the expression given in the problem:

\[ \lambda \log\left(\frac{x^a}{x^a + 1}\right) + C \]

### Step 8: Identifying Parameters
From the integral, we can identify:

- \(\lambda = 2\)
- \(a = 2\)

### Step 9: Calculating \(a + \lambda\)
Now we calculate:

\[ a + \lambda = 2 + 2 = 4 \]

### Final Answer
Thus, the value of \(a + \lambda\) is:

\[ \boxed{4} \]