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

In summary, the substitution caused the integral to become inaccurate. The original integral is \int \sin^2(x)dx=\frac{1}{2} \int 1-\cos(2x)dx= \frac{x}{2}-\frac{\sin(2x)}{4}.
  • #1
zee_22
5
0
Use the identity Sin[x]^2 = [1-Cos[2x]]/2
to help calculate integral from 0 to t Sin[x]^2 dx
This 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!:confused:
 
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  • #2
zee_22 said:
Use the identity Sin[x]^2 = [1-Cos[2x]]/2
to help calculate integral from 0 to t Sin[x]^2 dx
This 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!:confused:

Why did you make a substitution after you used the trig identity to get rid of the sine squared, cos(2x) is certainly integrable.
 
  • #3
back to the integration problem

d_leet said:
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
 
  • #4
zee_22 said:
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?
 
  • #5
Do NOT post homework questions or any other questions in the tutorials section! :grumpy:

Warning sent off to mods.
 
  • #6
arildno said:
Do NOT post homework questions or any other questions in the tutorials section! :grumpy:

Warning sent off to mods.
im very sorry ,I'm new ,i didn't know..
 
  • #7
zee_22 said:
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.

[tex]\sin(x)^2[/tex] and [tex]\sin(x^2)[/tex]
 
  • #8
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
 
  • #9
u rnt using [x] to denote gretest integer function? that would make it slightly complex
 
  • #10
No, he's not using [x] as the greatest integer function. Anyway, here's the original integral.

[tex]\int \sin^2(x)dx=\frac{1}{2} \int 1-\cos(2x)dx= \frac{x}{2}-\frac{\sin(2x)}{4}[/tex]

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

1. What is the purpose of integrating Sin^2[x] from 0 to t?

The purpose of integrating Sin^2[x] from 0 to t is to find the area underneath the curve of the function Sin^2[x] between the limits of 0 and t. This can be useful in solving various mathematical and scientific problems.

2. How do I solve an integral with the identity Sin^2[x]?

To solve an integral with the identity Sin^2[x], you can use the trigonometric identity Sin^2[x] = 1/2(1-cos2x). This will help simplify the integral and make it easier to solve.

3. What are the steps to integrating Sin^2[x] from 0 to t?

The steps to integrating Sin^2[x] from 0 to t are as follows:
1. Use the identity Sin^2[x] = 1/2(1-cos2x) to simplify the integral
2. Apply the power rule to integrate 1/2(1-cos2x)
3. Use the limits of 0 and t to evaluate the definite integral
4. Simplify the resulting expression to get the final answer.

4. Can I use a calculator to solve this integral?

Yes, you can use a calculator to solve the integral of Sin^2[x] from 0 to t. Most scientific calculators have built-in integration functions that can solve integrals with trigonometric functions.

5. What are some real-world applications of integrating Sin^2[x] from 0 to t?

Integrating Sin^2[x] from 0 to t has various applications in physics, engineering, and other scientific fields. For example, it can be used to calculate the displacement of a moving object with sinusoidal motion, or to calculate the work done by a variable force acting on an object. It can also be used in signal processing and electrical engineering to analyze signals with sinusoidal components.

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