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प्रश्न
Evaluate the following.
`int x^2 e^4x`dx
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उत्तर
Let I = `int x^2 e^4x`dx
`= x^2 int e^4x dx - int[d/dx (x^2) int e^4x dx]` dx
`= x^2 * e^4x/4 - int 2x * e^4x/4` dx
`= (x^2 * e^4x)/4 - 1/2 int x * e^4x` dx
`= (x^2 * e^4x)/4 - 1/2 [x int e^4x dx - int (d/dx (x) int e^4x dx) dx]`
`= (x^2 * e^4x)/4 - 1/2 [x * e^4x/4 - int 1 * e^4x/4 dx]`
`= (x^2 e^4x)/4 - 1/2[(x * e^4x)/4 - 1/4 int e^4x dx]`
`= (x^2 e^4x)/4 - 1/2[(x * e^4x)/4 - 1/4 * e^4x/4]` + c
`= (x^2 e^4x)/4 - (x e^4x)/8 + e^4x/32` + c
∴ I = `(e^4x)/4 [x^2 - x/2 + 1/8]` + c
Notes
The answer in the textbook is incorrect.
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