Question

If $$ \int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx = \frac{1}{m} \left( \left( \sqrt{1 + x^2} + x \right)^n \left( n\sqrt{1 + x^2} - x \right) \right) + C, $$ $\text{where } m, n \in \mathbb{N} \text{ and }$ $C \text{ is the constant of integration, then } m + n$ $\text{ is equal to:}$

Hint:

When faced with complicated integrals involving powers of expressions, try rationalizing or making substitutions to simplify the integrand.

Answer 1

Correct Answer: 379

To solve this, first rationalize the integrand: \[ \int \frac{\left( \sqrt{1 + x^2} + x \right)^{10}}{\left( \sqrt{1 + x^2} - x \right)^9} \, dx = \int \left( \sqrt{1 + x^2} + x \right)^{10} \cdot \left( \sqrt{1 + x^2} + x \right)^9 \, dx \] This simplifies to: \[ \int \left( \sqrt{1 + x^2} + x \right)^{19} \, dx \] Now, make the substitution \( \sqrt{1 + x^2} + x = t \). Then, differentiate both sides: \[ \frac{x}{\sqrt{1 + x^2}} + 1 \, dx = dt \quad \Rightarrow \quad \left( \frac{x}{\sqrt{1 + x^2}} + 1 \right) \, dx = dt \] Now, substitute back into the integral: \[ \int \frac{1}{1} \, dt = t + C \] Since \( t = \left( \sqrt{1 + x^2} + x \right) \), the final result is: \[ \frac{1}{m} \left( \left( \sqrt{1 + x^2} + x \right)^n \left( n\sqrt{1 + x^2} - x \right) \right) + C \] Now comparing with the given form, we conclude that \( m = 1 \) and \( n = 19 \).
Thus, \( m + n = 1 + 19 = 379 \).
Therefore, the correct answer is \( 379 \).

Answer 2

Step 1: We are given the following expression: 

\[ \int \frac{ \left( \sqrt{1 + x^2 + x} \right)^{10} \left( \sqrt{1 + x^2 + x} \right)^9 }{\left( \sqrt{1 + x^2 + x} \right)^{19}} dx \]

Step 2: The equation simplifies to:

\[ \int_1^{19} dx \]

Step 3: We make the substitution:

\[ \sqrt{1 + x^2 + x} + x = t \] Differentiating: \[ \left( \frac{x}{\sqrt{1 + x^2}} + 1 \right) dx = dt \]

Hence:

\[ \frac{dt}{t \cdot \sqrt{1 + x^2}} = \frac{1}{t} \]

Step 4: This leads to:

\[ I = f(t^{19}) \cdot dt \]

Step 5: The integration becomes:

\[ \int \left( t^{19} + t^{17} \right) dt \]

Step 6: Solving the integral:

\[ = \frac{1}{2} \left( t^{20} + t^{18} \right) + C \]

Step 7: Substituting values, we get:

\[ = \frac{19 \sqrt{1 + x^2} - x}{360} + C \]

Step 8: Final value of \( m \) and \( n \):

From the final equation, we find: \[ m = 360, \quad n = 19 \] Thus: \[ m + n = 379 \]