How do you find the antiderivative of #int (x^3cosx) dx#?

Answer 1

#x^3sinx+3x^2cosx-6xsinx-6cosx+C#

#I=intx^3cosxdx#
This will require multiple iterations of integration by parts (IBP). Integration by parts takes the form #intudv=uv-intvdu#.
So, for #intx^3cosxdx# as #intudv#, let:
#{(u=x^3),(dv=cosxdx):}#
Now, differentiating #u# and integrating #dv#, we see that:
#{(u=x^3,=>,du=3x^2dx),(dv=cosxdx,=>,v=sinx):}#

Now, plugging this into the IBP formula:

#I=uv-intvdu=x^3sinx-int3x^2sinxdx#
Now, for #int3x^2sinxdx#, perform IBP again:
#{(u=3x^2,=>,du=6xdx),(dv=sinxdx,=>,v=-cosx):}#

Thus:

#I=x^3sinx-[3x^2(-cosx)-int6x(-cosx)dx]#

Pay close attention to sign:

#I=x^3sinx+3x^2cosx-int6xcosxdx#

IBP again on the integral:

#{(u=6x,=>,du=6dx),(dv=cosxdx,=>,v=sinx):}#

So:

#I=x^3sinx+3x^2cosx-[6xsinx-int6sinxdx]#
Since #intsinxdx=-cosx#:
#I=x^3sinx+3x^2cosx-[6xsinx+6sinx]#
#I=x^3sinx+3x^2cosx-6xsinx-6cosx+C#
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Answer 2

To find the antiderivative of ( \int x^3 \cos(x) , dx ), you can use integration by parts. Let ( u = x^3 ) and ( dv = \cos(x) , dx ). Then, ( du = 3x^2 , dx ) and ( v = \sin(x) ). Apply the integration by parts formula:

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

Substitute the values:

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

Integrate ( \int \sin(x) \cdot 3x^2 , dx ) using integration by parts again or by using a simple substitution. Then, simplify the expression if necessary.

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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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