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प्रश्न
Evaluate the following.
`int x^3 e^(x^2)`dx
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उत्तर
Let I = `int x^3 e^(x^2)`dx
`= int x^2 * x * e^(x^2)` dx
Put x2 = t
∴ 2x . dx = dt
∴ x dx = `dt/2`
∴ I = `1/2 int te^t` dt
`= 1/2 [t int e^t dt - int [d/dt (t) int e^t dt] dt]`
`= 1/2 [te^t - int 1 * e^t dt]`
`= 1/2 (te^t - e^t) + c = 1/2 e^t (t - 1)` + c
∴ I = `1/2 e^(x^2) (x^2 - 1)` + c
Notes
The answer in the textbook is incorrect.
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