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What is the integral of #int x^3 cos(x^2) dx #?

Answer 1

Isaiah Allen

I found: #1/2x^2sin(x^2)+1/2cos(x^2)+c#

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Answer 2

Luke Alvarez

To find the integral of ( \int x^3 \cos(x^2) , dx ), you can use integration by parts. Let ( u = x^3 ) and ( dv = \cos(x^2) , dx ). Then, ( du = 3x^2 , dx ) and ( v = \int \cos(x^2) , dx ). Integrating ( \cos(x^2) ) with respect to ( x ) does not yield a simple elementary function, so ( v ) cannot be expressed in terms of elementary functions. However, you can still find the integral using the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substituting the values:

[ = x^3 \int \cos(x^2) , dx - \int \left(\int \cos(x^2) , dx \right) (3x^2) , dx ]

At this point, the integral ( \int \cos(x^2) , dx ) cannot be expressed in terms of elementary functions, so you cannot find a closed-form expression for the integral of ( \int x^3 \cos(x^2) , dx ).

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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.