If `int(x^(pq-p-1))/((x^p +1)^(q))dx= 2(1+x^(-p))^(1-q)/(lambda p(q-1))+c " "(p, q in N - {1})`, then the value of `lambda` is (here, c is an arbitary constant)

Answer

To solve the given integral and find the value of \( \lambda \), we start with the integral:

\[ I = \int \frac{x^{pq - p - 1}}{(x^p + 1)^q} \, dx \]

### Step 1: Rewrite the Integral
We can rewrite the integral by factoring out \( x^p \) from the denominator:

\[ I = \int \frac{x^{pq - p - 1}}{(x^p(1 + \frac{1}{x^p}))^q} \, dx = \int \frac{x^{pq - p - 1}}{x^{pq}(1 + \frac{1}{x^p})^q} \, dx \]

### Step 2: Simplify the Integral
This simplifies to:

\[ I = \int \frac{x^{pq - p - 1 - pq}}{(1 + \frac{1}{x^p})^q} \, dx = \int \frac{x^{-p - 1}}{(1 + \frac{1}{x^p})^q} \, dx \]

### Step 3: Substitution
Let \( t = 1 + x^{-p} \). Then, we differentiate:

\[ dt = -\frac{1}{p} x^{-p - 1} \, dx \implies dx = -p x^{p} dt \]

Substituting \( x^{-p} = t - 1 \) gives us:

\[ dx = -p (t - 1)^{-1/p} dt \]

### Step 4: Substitute in the Integral
Now we can substitute \( dx \) and the new variable \( t \) into the integral:

\[ I = \int \frac{-p (t - 1)^{-1/p}}{t^q} (-\frac{1}{p} x^{-p - 1} \, dx) = \int \frac{1}{t^q} dt \]

### Step 5: Solve the Integral
The integral becomes:

\[ I = -\frac{1}{p} \int t^{-q} dt = -\frac{1}{p} \cdot \frac{t^{1 - q}}{1 - q} + C \]

Re-substituting \( t = 1 + x^{-p} \):

\[ I = -\frac{1}{p(1 - q)} \left(1 + x^{-p}\right)^{1 - q} + C \]

### Step 6: Equate to Given Expression
We are given that:

\[ I = \frac{2(1 + x^{-p})^{1 - q}}{\lambda p (q - 1)} + C \]

Equating the two expressions for \( I \):

\[ -\frac{1}{p(1 - q)} \left(1 + x^{-p}\right)^{1 - q} = \frac{2(1 + x^{-p})^{1 - q}}{\lambda p (q - 1)} \]

### Step 7: Solve for \( \lambda \)
Cancelling \( (1 + x^{-p})^{1 - q} \) from both sides (valid since \( p, q \in \mathbb{N} \setminus \{1\} \)):

\[ -\frac{1}{p(1 - q)} = \frac{2}{\lambda p (q - 1)} \]

Cross-multiplying gives:

\[ -\lambda = 2(1 - q) \]

Thus, we find:

\[ \lambda = -2(1 - q) \]

### Final Answer
The value of \( \lambda \) is:

\[ \lambda = 2(q - 1) \]