Integrating Sin^2[x] from 0 to t: Solve with Identity

[zee_22]

Use the identity Sin[x]^2 = [1-Cos[2x]]/2
to help calculate integral from 0 to t Sin[x]^2 dxThis question seems really easy but I am having some difficulty with it.
This is what I am thinking :
first put the [1-Cos[2x]]/2 instead of the Sin[x]^2
so this is what i have now (int from 0 to t) [1-Cos[2x]]/2dx
then i thought let u = cos[2x] then du = - Sin[2x]dx
and when x= 0 then u = 1 and when x= t then u= cos[2t]
then (int from 0 to t) [1-Cos[2x]]/2dx
=-1/2 (int from 1 to cos[2t] ) [1-u]/[Sin[2x]du]
Now i don't know what to do please help!

 

[d_leet]

Use the identity Sin[x]^2 = [1-Cos[2x]]/2
to help calculate integral from 0 to t Sin[x]^2 dxThis question seems really easy but I am having some difficulty with it.
This is what I am thinking :
first put the [1-Cos[2x]]/2 instead of the Sin[x]^2
so this is what i have now (int from 0 to t) [1-Cos[2x]]/2dx
then i thought let u = cos[2x] then du = - Sin[2x]dx
and when x= 0 then u = 1 and when x= t then u= cos[2t]
then (int from 0 to t) [1-Cos[2x]]/2dx
=-1/2 (int from 1 to cos[2t] ) [1-u]/[Sin[2x]du]
Now i don't know what to do please help!

Why did you make a substitution after you used the trig identity to get rid of the sine squared, cos(2x) is certainly integrable.

 

[zee_22]

back to the integration problem

Why did you make a substitution after you used the trig identity to get rid of the sine squared, cos(2x) is certainly integrable.

the substitution was the first thing i did i should leave the sin[x]^2 like it is then??
let sin[x]= u ?
what is the integral of Cos[2x]??
please

 

[zee_22]

the substitution was the first thing i did i should leave the sin[x]^2 like it is then??
let sin[x]= u ?
what is the integral of Cos[2x]??
please

Ok i think i got it:shy:
I just one clerification is sin^2[x] the same as Sin[x]^2?

 

[arildno]

Do NOT post homework questions or any other questions in the tutorials section!

Warning sent off to mods.

 

[zee_22]

Do NOT post homework questions or any other questions in the tutorials section!

Warning sent off to mods.

im very sorry ,I'm new ,i didn't know..

 

[Jameson]

Ok i think i got it:shy:
I just one clerification is sin^2[x] the same as Sin[x]^2?

Yes that's correct. That notation is used to not confuse the following expressions.

$$\sin(x)^2$$ and $$\sin(x^2)$$

 

[aiglet_2000]

not much trouble by the way
but a slight mistake
if u=cos2x
than d(u)=-sin2xd(2x)
=-2sin2xdx
but i don't really know why u r making this substitution?
cos2x is a basic integration, but if u want to do it this way which is useless than do tell i will tell u than how to do this useless thing

 

[aiglet_2000]

u rnt using [x] to denote gretest integer function? that would make it slightly complex

 

[Jameson]

No, he's not using [x] as the greatest integer function. Anyway, here's the original integral.

$$\int \sin^2(x)dx=\frac{1}{2} \int 1-\cos(2x)dx= \frac{x}{2}-\frac{\sin(2x)}{4}$$

No constant of integration because it's a definite integral. Just plug in your bounds.