


Treatment of x2 in curve integral: Why can the standard answer be ignored (1/3) x3?
Apr 19, 2025 pm 08:06 PMDetailed explanation of the curve integral example problem: Clever use of the element exchange method
This article provides detailed answers to questions about x2 processing in the integral step in a curve integral example problem. In the example, the process of calculating ∫x2sin(x3)dx, the standard answer is confusing: the standard answer seems to directly transform x2dx into (1/3)dx3, which is different from the conventional integral method.
This processing method does not ignore the integral result of x2 (1/3) x3, but cleverly applies the exchange integral method .
Detailed analysis:
To calculate ∫x2sin(x3)dx, we use the u substitution method:
Let u = x3, then du = 3x2dx. Therefore, x2dx = (1/3)du.
Bring the substitution into the original points:
∫x2sin(x3)dx = ∫sin(u)(1/3)du = (1/3)∫sin(u)du = -(1/3)cos(u) C
Finally, replace u = x3 and get the final result: -(1/3)cos(x3) C.
in conclusion:
The standard answer does not ignore the integral of x2, but converts x2dx into (1/3)du through the conversion integral method, thereby simplifying the integration process and finally getting the correct result. Therefore, the previous questions stem from the deviation in understanding the exchange element integral method.
The above is the detailed content of Treatment of x2 in curve integral: Why can the standard answer be ignored (1/3) x3?. For more information, please follow other related articles on the PHP Chinese website!

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