The correct option is (a) \(\frac{2}{n(n+1)} e^{\frac{n(n+1)}{2}x} \left [x^3+3x^2 [\frac{2}{n(n+1)}]^1+6x[\frac{2}{n(n+1)}]^2 +6[\frac{2}{n(n+1)}]^3\right ]\)
Easy explanation: Add constant automatically
By, f(x)=\(\int uvdx=\sum_{i=0}^n (-1)^i u_i v^{i+1}\)
Let, u = x^3 and v=e^x e^2x e^3x…..e^nx=e^x(1+2+3+…n)=\(e^{\frac{n(n+1)x}{2}}\),
\(\int x^3 e^x e^2x e^3x……..e^nx dx\)
\(=x^3 \frac{2}{n(n+1)} e^{\frac{n(n+1)}{2}x}+3x^2 [\frac{2}{n(n+1)}]^2 e^{\frac{n(n+1)}{2}x}\) \(+6x[\frac{2}{n(n+1)}]^3 e^{\frac{n(n+1)}{2}x}+6[\frac{2}{n(n+1)}]^4 e^{\frac{n(n+1)}{2}x}\)
=\(\frac{2}{n(n+1)} e^{\frac{n(n+1)}{2}x} \left [x^3+3x^2 [\frac{2}{n(n+1)}]^1+6x[\frac{2}{n(n+1)}]^2+6[\frac{2}{n(n+1)}]^3\right]\)