If ∫ (√(1+x^2)+x)^(10)/(√(1+x^2)-x)^9 dx = 1/m (((√1+x^2+x)^n(n(√(1+x^2)-x)) + C where C is the constant of integration and m, n ∈ N,

Question

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

Answer 1

Answer is: 379 

Rationalize

\( \Rightarrow \int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} \times \frac{\left(\sqrt{1+x^2}+x\right)^9}{\left(\sqrt{1+x^2}+x\right)^9} d x \)

\(\Rightarrow \int \frac{\left(\sqrt{1+x^2}+x\right)^{19}}{1} d x \)

Put \(\sqrt{1+\mathrm{x}^2}+\mathrm{x}=\mathrm{t}\)

\( \left(\frac{\mathrm{x}}{\sqrt{1+\mathrm{x}^2}}+1\right) \mathrm{dx}=\mathrm{dt} \)

\( \mathrm{dx}=\frac{\mathrm{dt}}{\mathrm{t}} \sqrt{1+\mathrm{x}^2}\)

Now as \(\sqrt{1+\mathrm{x}^2}+\mathrm{x}=\mathrm{t}\)

so \(\sqrt{1+\mathrm{x}^2}-\mathrm{x}=\frac{1}{\mathrm{t}}\)

\(\therefore \sqrt{1+\mathrm{x}^2}=\frac{1}{2}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)\)

Thus \(\mathrm{I}=\int \mathrm{t}^{19} \cdot \frac{\mathrm{dt}}{\mathrm{t}} \cdot \frac{1}{2}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)\)

\( \Rightarrow \frac{1}{2} \int\left(\mathrm{t}^{1^{19}}+\mathrm{t}^{17}\right) \mathrm{dt} \)

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

\(=\frac{\mathrm{t}^{19}}{360}\left[9 \mathrm{t}+\frac{10}{\mathrm{t}}\right]+\mathrm{C} \)

\( =\frac{\mathrm{t}^{19}}{360}\left[9\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)+\frac{1}{\mathrm{t}}\right]+\mathrm{C} \)

\( \Rightarrow \frac{\left(\sqrt{1+\mathrm{x}^2}+\mathrm{x}\right)^{19}}{360}\left[9\left(2 \sqrt{1+\mathrm{x}^2}\right)+\left(\sqrt{1+\mathrm{x}^2}-\mathrm{x}\right)\right]+\mathrm{C} \)

\( \Rightarrow \frac{\left(\sqrt{1+\mathrm{x}^2}+\mathrm{x}\right)^{19}}{360}\left[19 \sqrt{1+\mathrm{x}^2}-\mathrm{x}\right]+\mathrm{C} \)

\( \therefore \mathrm{~m}=360, \mathrm{n}=19 \)

\(\therefore \mathrm{~m}+\mathrm{n}=379\)