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int(x-3e^(3x))dx

(x3e3x)dx \int\left(x-3 e^{3 x}\right) d x

Full solution

Q. (x3e3x)dx \int\left(x-3 e^{3 x}\right) d x
  1. Separate into two parts: Separate the integral into two parts.\newline(x3e3x)dx=xdx3e3xdx\int(x - 3e^{3x})dx = \int xdx - \int 3e^{3x}dx
  2. Integrate xx: Integrate xdx\int x\,dx.xdx=12x2\int x\,dx = \frac{1}{2}x^2
  3. Integrate 3e3x3e^{3x}: Integrate 3e3xdx\int 3e^{3x}\,dx. Let u=3xu = 3x, then dudx=3\frac{du}{dx} = 3, dx=du3dx = \frac{du}{3}. 3e3xdx=eudu=euC\int 3e^{3x}\,dx = \int e^u \,du = \frac{e^u}{C}, where CC is 13\frac{1}{3}.