(√1+x^2+x)^10(√1+x^2-x)^9,dx = 1m[(√1+x^2+x)^n(n√1+x^2-x)]+C, where C is the constant of i…

Question

$\displaystyle\int\frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^{9}}\,dx = \dfrac1m\left[(\sqrt{1+x^2}+x)^n\big(n\sqrt{1+x^2}-x\big)\right]+C$, where $C$ is the constant of integration and $m,n\in\mathbb{N}$. Then $m+n$ is equal to \_\_\_\_\_\_\_\_\_.

Answer: 379

Accepted Answer

Since $(\sqrt{1+x^2}+x)(\sqrt{1+x^2}-x)=1$, the integrand equals $(\sqrt{1+x^2}+x)^{19}$. With the substitution $t=\sqrt{1+x^2}+x=\sec\theta+\tan\theta$ (so $\sqrt{1+x^2}=\frac{t^2+1}{2t}$, $x=\frac{t^2-1}{2t}$), the integral works out, after simplification, to match the given form with

$$n=19,\qquad m=360$$

$m+n = 379$

Solutions · 1