How do you integrate #int x^2 cos3 x dx # using integration by parts?

Answer 1

#I = (9x^2sin(3x) +18xcos(3x) -2sin(3x))/9 + c#

#I = intx^2cos(3x)dx#
Say #u = x^2# so #du = 2x# and #dv = cos(3x)# so #v = sin(3x)/3#
#I = (x^2sin(3x))/3 - 2/3intxsin(3x)dx#
Say #u = x# so #du = 1# and #dv = sin(3x)# so #v = -cos(3x)/3#
#I = (x^2sin(3x))/3 - 2/3(-xcos(3x) +1/3intcos(3x)dx)#
#I = (x^2sin(3x))/3 - 2/3(-xcos(3x) +sin(3x)/9) + c#
#I = (9x^2sin(3x) +18xcos(3x) -2sin(3x))/9 + c#
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Answer 2

To integrate ( \int x^2 \cos(3x) , dx ) using integration by parts, follow these steps:

  1. Choose parts: Let ( u = x^2 ) and ( dv = \cos(3x) , dx ).
  2. Compute differentials: Calculate ( du ) and ( v ) by differentiating ( u ) and integrating ( dv ).
  3. Apply integration by parts formula: [ \int u , dv = uv - \int v , du ].
  4. Substitute values and integrate: Plug in the values for ( u ), ( du ), ( v ), and ( dv ), then integrate the remaining term.

Applying these steps, the integral becomes:

[ \int x^2 \cos(3x) , dx = \frac{x^2 \sin(3x)}{3} - \frac{2}{3} \int x \sin(3x) , dx ]

The remaining integral ( \int x \sin(3x) , dx ) may require further integration by parts or another method depending on your preferences and the context of the problem.

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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.

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.

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.

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.

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