How do you integrate #int x^3 ln x^2 dx # using integration by parts?
Now, to this end, we will use the Rule of I ntegration byParts arts :
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To integrate ∫x^3 ln(x^2) dx using integration by parts, follow these steps:

Choose u and dv: Let u = ln(x^2) and dv = x^3 dx.

Find du and v: Differentiate u to find du, and integrate dv to find v.
du = (1/x^2) * 2x dx = 2/x dx
v = ∫x^3 dx = (1/4)x^4

Apply the integration by parts formula: ∫u dv = uv  ∫v du
∫x^3 ln(x^2) dx = (ln(x^2)) * (1/4)x^4  ∫(1/4)x^4 * (2/x) dx

Simplify and integrate the remaining integral: ∫(1/4)x^4 * (2/x) dx = (1/2) ∫x^3 dx = (1/2) * (1/4)x^4 = (1/8)x^4

Combine the results: ∫x^3 ln(x^2) dx = (1/4)x^4 ln(x^2)  (1/8)x^4 + C
Therefore, the integral of x^3 ln(x^2) dx using integration by parts is (1/4)x^4 ln(x^2)  (1/8)x^4 + C, where C is the constant of integration.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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