How do you integrate #int x^3 sec^2 x dx # using integration by parts?
The first two steps are OK. After that you need polylogarithmic functions (beyond the scope of an introductory course).
So, we have
So we have
The last integral involves the polylogarithmic function.
I don't know enough about that to explain it. (Good luck.)
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To integrate ∫x^3 sec^2(x) dx using integration by parts, you can follow these steps:

Choose u and dv: Let u = x^3 (function to differentiate) Let dv = sec^2(x) dx (function to integrate)

Find du and v: Calculate du/dx = 3x^2 (derivative of u with respect to x) Integrate dv to get v: ∫sec^2(x) dx = tan(x)

Apply the integration by parts formula: ∫u dv = uv  ∫v du

Substitute into the formula: ∫x^3 sec^2(x) dx = x^3 tan(x)  ∫tan(x) * 3x^2 dx

Integrate the remaining integral: ∫tan(x) * 3x^2 dx can be further integrated using substitution or integration by parts if needed.

Simplify the result and add the constant of integration as necessary.
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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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