Suppose F(x) = the integral of: sin (t^2+1) as x goes from x^2 to e^(2x) dt
Evaluate F'(x).

Question

Suppose F(x) = the integral of: sin (t^2+1) as x goes from x^2 to e^(2x)  dt 
Evaluate F'(x).

anonymous · Answer

$$F(x)=\int_{x^2}^{e^{2x}}sin(t^2+1) dt$$?

anonymous · Answer

i think gimmick is to break it up into two pieces. for example if you want the derivative of 
$$F(x)=\int_0^{e^{2x}}\sin(t^2+1)dt$$ by the fundamental theorem and the chain rule you get 
$$F'(x)=\sin(e^{4x}+1)2e^{2x}$$

anonymous · Answer

and 
$$F(x)=\int_{x^2}^0 sin(t^2+1)dt=-\int_0^{x^2}sin(t^2+1)dt$$\]

anonymous · Answer

whose derivative is 
$$F'(x)=-\sin(x^2+1)2x$$

anonymous · Answer

oh look!  i see trees!

myininaya · Answer

my thingy got messed up

anonymous · Answer

$$\color{green}{\text{trees are green}}$$

anonymous · Answer

$$\color{green}{\text{sorry about your thingy}}$$

myininaya · Answer

$$F(x)=G(e^{2x})-G(x^2) $$
where G represents the antiderivative of $$g(t)=\sin(t^2+1)$$
$$F'(x)=(e^{2x})'g(e^{2x})-(x^2)'g(x^2)$$

anonymous · Answer

answer is 
$$F'(x)=2e^{2x}\sin(e^{4x}+1)-2x\sin(x^4+1)$$

myininaya · Answer

right!

myininaya · Answer

i c no trees here

anonymous · Answer

$$\color{#FF0099}{\text{time for you to get an icon of some sort}}$$

anonymous · Answer

$$\color{#00FF11}{\text{no self respecting superhero should be a chair with feet}}$$

anonymous · Answer

$$\color{red}{\text{btw sorted mathmom for you while you went to eat}}$$