10. Find (d^(2))/(d x^(2))\int ^x_0\left(\int ^(\sin t)_1\sqrt{1+u^(4)}du\right)dt\text{.}

Question

10. Find
`(d^(2))/(d x^(2))\int ^x_0\left(\int ^(\sin t)_1\sqrt{1+u^(4)}du\right)dt\text{.}`

Answer 1

Let g(t) denote the inner integral. By the fundamental theorem of calculus, the first derivative is

`\displaystyle (d)/(dx) \int_0^x g(t) \, dt = g(x)`

Then using the FTC again, differentiating g gives

`\displaystyle (dg)/(dx) = (d)/(dx) \int_1^(\sin(x)) √(1+u^4) \, du = \boxed{\cos(x) √(1+\sin^4(x))}`