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प्रश्न
Evaluate the following.
`int x^2 *e^(3x)`dx
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उत्तर
Let I = `int x^2 e^(3x)`dx
`= x^2 int e^(3x) dx - int[d/dx (x^2) int e^(3x) dx]` dx
`= x^2 * (e^(3x)/3) - int 2x * e^(3x)/3` dx
`= (x^2)/3 e^(3x) - 2/3 int x * e^(3x)` dx
`= (x^2)/3 e^(3x) - 2/3 [x int e^(3x) dx - int (d/dx (x) int e^(3x) dx) dx]`
`= (x^2 * e^(3x))/3 - 2/3 [x * e^(3x)/3 - int 1 * e^(3x)/3 dx]`
`= (x^2 * e^(3x))/3 - 2/3 [1/3 xe^(3x) - 1/3 int e^(3x) dx]`
`= (x^2 * e^(3x))/3 - 2/3 [1/3 xe^(3x) - 1/3 * e^(3x)/3]` + c
∴ I = `1/3 x^2 * e^(3x) - 2/9 xe^(3x) + 2/27 e^(3x) + c`
Notes
The answer in the textbook is incorrect.
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