integrate by parts--  x^2.e^x^3 (explain too)  wrt dxits x squared multiplied by e raised to the power x cube

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You first need to make the following substitution such that:

`x^3 = t => 3x^2 dx = dt => x^2 dx = (dt)/3`

You need to change variable to integrand such that:

`int x^2*e^(x^3)dx = int e^t*(dt)/3 `

Notice that you do not need to integrate by parts since you may integrate using the following formula `int e^x dx = e^x + c` .

Reasoning by analogy yields:

`(1/3) int e^t dt = (1/3)e^t + c`

Substituting back `x^3`  for t yields:

`int x^2*e^(x^3)dx = (e^(x^3))/3 + c`

Hence, evaluating the given integral using substitution and not integration by parts, yields `int x^2*e^(x^3)dx = (e^(x^3))/3 + c.`