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How do you use substitution to integrate # xsin x^2 dx# from [0,pi]?

Answer 1

Evelyn Agard

I found #int_0^pixsin(x^2)dx=0.951#

Try this:

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Answer 2

Emily Ammerman

To integrate (x\sin(x^2) , dx) from (0) to (\pi) using substitution, follow these steps:

Let (u = x^2), then (du = 2x , dx).
Solve for (x , dx) in terms of (du): (x , dx = \frac{du}{2}).
Substitute (u = x^2) and (x , dx = \frac{du}{2}) into the integral.
The integral becomes (\frac{1}{2} \int \sin(u) , du).
Integrate (\sin(u)) with respect to (u): (-\frac{1}{2}\cos(u)).
Substitute back (u = x^2): (-\frac{1}{2}\cos(x^2)).
Evaluate the expression at the upper and lower limits of integration, which are (\pi) and (0), respectively.
Plug in (\pi): (-\frac{1}{2}\cos(\pi^2)).
Plug in (0): (-\frac{1}{2}\cos(0^2)).
Calculate the values: (-\frac{1}{2}\cos(\pi^2) + \frac{1}{2}\cos(0^2)).
Simplify the expression if needed.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.