Treatment of x² in curve integral: Why can the standard answer be ignored (1/3) x³?

Detailed 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.

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