Solution to an equation involving an integral I am stuck at how to how  solve the following equation involving an integral
$$ \int_0^t \sqrt{1+\cos^2x} \ dx=a,$$ where $a$ is some constant. Even if we take $a$ to be some numerical constant ,say,3,I have no idea how to solve it numerically. Any suggestions\hints\links are greatly appreciated.

You can use Newton's method, combined with a quadrature formula. You want to compute a zero of $F(t)= \int_0^t\sqrt{1+\cos^2 x} dx -a$. So, take an initial approximation $t_0$ and form the sequence
$$
t_{n+1} = t_n-\dfrac{F(t_n)}{F'(t_n)}= t_n-\dfrac{\int_0^{t_n} \sqrt{1+\cos^2 x}dx -a}{\sqrt{1+\cos^2 t_n}}\approx t_n-\dfrac{v_n-a}{\sqrt{1+\cos^2 t_n}},
$$
where $v_n$ is a quadrature rule of your choice (Trapezoidal, Simpson, etc.)
$$
v_n = \sum_{i=0}^k \omega_{i,n} \sqrt{1+\cos^2 \xi_{i,n}}.
$$
