Let I = ∫x2ex dx
Taking \(x^2\) as first function and ex as second function and integrating by parts, we obtain
I= x2∫ex dx - ∫{\((\frac {d}{dx}x^2)\)∫exdx} dx
I = x2ex - ∫2x.exdx
I = x2ex-2∫x.ex dx
Again integrating by parts,we obtain
I =x2ex - 2[x.∫exdx - ∫{\((\frac {d}{dx}x)\).∫exdx} dx]
I = x2ex - 2[xex-∫exdx]
I = x2ex - 2[xex - ex]
I = x2ex - 2xex + 2ex + C
I = ex(x2 - 2x + 2) + C
If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:
The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,