233k views
1 vote
10. Find
(d^(2))/(d x^(2))\int ^x_0\left(\int ^(\sin t)_1\sqrt{1+u^(4)}du\right)dt\text{.}

User Snakehiss
by
8.4k points

1 Answer

5 votes

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))}

User DZN
by
8.6k points

Related questions

1 answer
1 vote
214k views
1 answer
0 votes
43.4k views
asked May 5, 2021 226k views
Nulle asked May 5, 2021
by Nulle
8.3k points
2 answers
1 vote
226k views