`int((sqrt(x))^5dx)/((sqrt(x))^7+x^6)=lambdalog((x^a)/(x^a+1))+c,` then `a+lambda`

Options

A. `=2`
B. `gt 2`
C. `lt2`
D. `=1`

Explanation

To solve the integral

\[ \int \frac{(\sqrt{x})^5}{(\sqrt{x})^7 + x^6} \, dx = \lambda \log\left(\frac{x^a}{x^a + 1}\right) + c, \]

we will follow these steps:

### Step 1: Rewrite the integral
We start by rewriting the integral in a more manageable form. We know that \(\sqrt{x} = x^{1/2}\), so we can express the integral as:

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

### Step 2: Factor out common terms in the denominator
Next, we can factor \(x^{7/2}\) out of the denominator:

\[ \int \frac{x^{5/2}}{x^{7/2}(1 + x^{6 - 7/2})} \, dx = \int \frac{x^{5/2}}{x^{7/2}(1 + x^{3/2})} \, dx. \]

This simplifies to:

\[ \int \frac{1}{x^{1/2}(1 + x^{3/2})} \, dx. \]

### Step 3: Substitute variables
Now we can make a substitution to simplify the integral. Let:

\[ t = x^{3/2} \implies dt = \frac{3}{2} x^{1/2} \, dx \implies dx = \frac{2}{3} t^{-1/3} \, dt. \]

### Step 4: Change the variable in the integral
Substituting \(x\) and \(dx\) into the integral gives:

\[ \int \frac{1}{x^{1/2}(1 + t)} \cdot \frac{2}{3} t^{-1/3} \, dt. \]

Since \(x = t^{2/3}\):

\[ x^{1/2} = (t^{2/3})^{1/2} = t^{1/3}, \]

the integral becomes:

\[ \int \frac{2}{3} \cdot \frac{t^{-1/3}}{t^{1/3}(1 + t)} \, dt = \frac{2}{3} \int \frac{1}{1 + t} \, dt. \]

### Step 5: Integrate
The integral of \(\frac{1}{1 + t}\) is:

\[ \int \frac{1}{1 + t} \, dt = \log(1 + t) + C. \]

Thus, we have:

\[ \frac{2}{3} \log(1 + t) + C. \]

### Step 6: Substitute back for \(t\)
Now substitute back \(t = x^{3/2}\):

\[ \frac{2}{3} \log(1 + x^{3/2}) + C. \]

### Step 7: Compare with the given form
We need to compare this result with the given form:

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

### Step 8: Identify \(\lambda\) and \(a\)
From our result, we can see that:

\[ \lambda = \frac{2}{3}, \quad a = \frac{3}{2}. \]

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

\[ a + \lambda = \frac{3}{2} + \frac{2}{3}. \]

To add these fractions, we need a common denominator:

\[ \frac{3}{2} = \frac{9}{6}, \quad \frac{2}{3} = \frac{4}{6}. \]

Thus,

\[ a + \lambda = \frac{9}{6} + \frac{4}{6} = \frac{13}{6}. \]

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

\[ \frac{13}{6}. \]