I was given the task to find the derivative of the following function:
$$F(x)=\int_0^ {g(x)}{\dfrac{1}{1+\sin^2(x)}dx}$$
With $$g(x) = \int_0^ {x^2}{\dfrac{1}{1+\sin^2(x)}dx}$$
My attempt was to apply the principal of integral calculations so that I know $F(x)$ without any integral signs. Then, I could find the derivative of that function. Sadly I am totally stuck at integrating
$$\int{\dfrac{1}{1+\sin^2(x)}dx}$$
Could someone help me with this? Also it would be great if someone could tell me if there is an easier way of integrating an expression like given above because “finding the derivative of an integral” sounds way more easy.
Greetings, Finn
Just use Leibniz' rule of differentiating under the integral sign to get: $$f'(x) =\frac1{1+\sin^2 g(x) } g'(x) $$ with $$g'(x) = \frac1{1+\sin^2 x^2}(2x)$$