Question

Find dy/dx for y = ∫(1+ sin^2u)/(1 + cos^2u)
interval: [π, π + x]

Answer 1

Fundamental Theorem of Calculus problem

Answer 2

Let \[g(x)=\frac{ 1+\sin^2x }{ 1+\cos^2x }\]
and define G(x) in which\[G'(x)=g(x)\]

Answer 3

Then the problem becomes\[\int\limits_{x}^{\pi+x}g(u)du=f(x)\]

Answer 4

Using our definition of G(x),\[\int\limits_{x}^{\pi+x}g(u)du=G(\pi+x)-G(x)=f(x)\]

Answer 5

We have to find f'(x), so...\[f'(x)=(G(\pi+x)-G(x))'=g(\pi+x)-g(x)\]

Answer 6

You can do the rest since we defined\[g(x)=\frac{ 1+\sin^2x }{ 1+\cos^2x }\]

Answer 7

From this problem, we can derive the formula that if\[f(x)=\int\limits_{a}^{b}g(x)dx\]
then
\[f'(x)=g(b)-g(a)\]