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The geometric series converges because the absolute value of its common ratio is below one. Its sum is

The terms form an arithmetic progression with common difference $2$. There are $(55-5)/2+1=26$ terms. The arithmetic series formula gives

The Leibniz integral rule combines contributions from both moving endpoints with differentiation under the integral sign:The partial derivative in the last integral holds $t$ fixed. Suitable regularity and convergence are assumed as stipulated.

Apply the Leibniz rule for the $n$th derivative of a product. Only the first three terms survive, because all derivatives of $x^2$ of order greater than two vanish and every derivative of $e^x$ is $e^x$:At $n=100$ this gives

On an interval where $\tan x$ is defined, use the substitution rule $t=\tan x$. Then $dx=dt/(1+t^2)$ and ${\cos }^{2}x=1/(1+t^2)$, givingThe inverse tangent expression is valid on each interval $(j\pi -\pi /2,j\pi +\pi /2)$ separately. The original integrand is continuous for all real $x$, since its denominator is at least $1$. To obtain one continuous antiderivative on the whole real axis, add $j\pi /2$ on the $j$th such interval and use the continuous limits at their endpoints.