What is the integral of (sin(x^2)*x)dx Thanks!

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nick-teal | High School Teacher | (Level 3) Adjunct Educator

Posted on

This is a classic u substitution problem.  The goal of a u sub problem is to make an integral easier by substituting u in for something we do not know how to integrate.

When doing a u sub problem we want to find the "ugly" part of the problem.  In this case it is the `x^2` that is inside the sin.

So we let `u = x^2`

Then we substitute in our u value into the integral and get

`int sin(u)xdx`

We are not done substituting yet though.  We still need to get rid of that x, and change dx into du.

To get dx, we take the derivative of our substitution, 

`u = x^2` 

`du = 2xdx` 

At this point we should recognize that we have xdx left in our integral, and that we have an 2xdx in the derivative of the substitution.  If we move the 2 to the other side we have 

`du* 1/2 = xdx`

Then we substitute 1/2 *du in for xdx in our integral and we are left with.

`int sin(u) (1/2) du`

This is much easier to integrate.  Trig rules tell us the integral of sin is -cos, so our integral equals


But we are not done quite yet! we still have to sub x^2 back in for the u.

After doing so, our answer to this integral problem becomes


zaurenwhilst's profile pic

zaurenwhilst | Student, Undergraduate | eNotes Newbie

Posted on


using change of variable(a property)

We let `x=t `

differentiating both sides we have

  `2dx=dt `

  `dx=dt/2 `

and replacing in above we have:



the integral of sin is -cos so w'll have:

`-1/2cost^2 `

but t = x so we finally have:

`-1/2cosx^2 `

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messian | eNotes Newbie

Posted on

Substitite x^2 by t. So that 2x.dx=dt. 

so the integral becomes 1/2 sin t dt, which is

-cos t + C, that is - cos x^2 + C.

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