Question

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

Answer 1

379

Answer 2

Here's how to solve this integral problem:

1. Simplify the integrand: Let $t = \sqrt{1+x^2}+x$. Use the property $(\sqrt{1+x^2}-x)(\sqrt{1+x^2}+x)=1$ to express the denominator in terms of $t$. The integral simplifies.

2. Express $dx$ in terms of $t$ and $dt$: Differentiate $t$ with respect to $x$ to find $\frac{dt}{dx}$. Then, express $\sqrt{1+x^2}$ in terms of $t$ by adding $t$ and $1/t$. Substitute this into the expression for $dx$.

3. Perform the integration: Substitute $dx$ to integrate with respect to $t$.

4. Match with the given form (Alternative and more robust approach): Instead of trying to manipulate the integrated expression to match the given form, differentiate the given form of the result with respect to $x$.

   • Let $F(x) = \frac{1}{m}((\sqrt{1+x^2}+x)^n(n\sqrt{1+x^2}-x))$.

   • Convert $F(x)$ entirely in terms of $t = \sqrt{1+x^2}+x$ and its reciprocal $1/t$.

   • Differentiate $F(x)$ using the chain rule: $\frac{dF}{dx} = \frac{dF}{dt} \cdot \frac{dt}{dx}$.

   • Calculate $\frac{dF}{dt}$ and $\frac{dt}{dx}$.

5. Compare and solve for m and n: Equate the derived $\frac{dF}{dx}$ with the original integrand. This directly gives $n$ and $m$.

6. Calculate m+n:

Following these steps, we find that $m+n = 379$.