please help me these two problems

[oceanc]

1. how do i compute the integral of sin2x/(1+(cosx)^2) ?
2. evaluate the limit: lim(sin(x))^x --- as x approches 0 from the right

 

[DrPhil]

1. how do i compute the integral of sin2x/(1+(cosx)^2) ?
2. evaluate the limit: lim(sin(x))^x --- as x approches 0 from the right

Without seeing your work, we have to assume you don't know how to start.

1. The numerator has a double angle in it. Can you make the denominator a function of 2x?

2. Do you know the value of 0^0? Can you prove it?

 

[oceanc]

Without seeing your work, we have to assume you don't know how to start.

1. The numerator has a double angle in it. Can you make the denominator a function of 2x?

2. Do you know the value of 0^0? Can you prove it?

for the first question, what should I set into U?
for the second, no...

 

[DrPhil]

for the first question, what should I set into U?
for the second, no...

Needs more effort, and a lot more thought.

1. Look up double-angle formulas. What can you make \(\displaystyle \cos^2{x} \) into?

2. the limit of \(\displaystyle \sin{x} \) as \(\displaystyle x\to 0 \) is \(\displaystyle x\). Would it be easier to deal with \(\displaystyle x^x\), instead of \(\displaystyle (\sin{x})^x\)?

 

[oceanc]

Needs more effort, and a lot more thought.

1. Look up double-angle formulas. What can you make \(\displaystyle \cos^2{x} \) into?

2. the limit of \(\displaystyle \sin{x} \) as \(\displaystyle x\to 0 \) is \(\displaystyle x\). Would it be easier to deal with \(\displaystyle x^x\), instead of \(\displaystyle (\sin{x})^x\)?

I tried setting cosx to U, then it would be the integral of sin2x.1/1+u^2
u'=-sinx, so if i could(and i couldn't) find the connection between -sinx and sin2x then i know how to solve this.

I also tried setting (cosx)^2 to U but it doesn't seem right...

 

[DrPhil]

I tried setting cosx to U, then it would be the integral of sin2x.1/1+u^2
u'=-sinx, so if i could(and i couldn't) find the connection between -sinx and sin2x then i know how to solve this.

I also tried setting (cosx)^2 to U but it doesn't seem right...

Since the numerator is sin(2x), it would be very handy if the derivative of your substitution was
....\(\displaystyle du = (\sin{2x})\ dx\)

For that to be true, \(\displaystyle u\) would have to be a function of \(\displaystyle \cos{2x}\). For the third time, look up the double angle formulas - can you make the denominator a function of \(\displaystyle \cos{2x}\)?