∫ex(2−x2) dx\int {e}^{x}(2-{x}^{2}) \, dx
1
Expand.
∫2ex−exx2 dx\int 2{e}^{x}-{e}^{x}{x}^{2} \, dx

2
Use Sum Rule: ∫f(x)+g(x) dx=∫f(x) dx+∫g(x) dx\int f(x)+g(x) \, dx=\int f(x) \, dx+\int g(x) \, dx.
∫2ex dx−∫exx2 dx\int 2{e}^{x} \, dx-\int {e}^{x}{x}^{2} \, dx

3
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
2∫ex dx−∫exx2 dx2\int {e}^{x} \, dx-\int {e}^{x}{x}^{2} \, dx

4
The integral of ex{e}^{x} is ex{e}^{x}.
2ex−∫exx2 dx2{e}^{x}-\int {e}^{x}{x}^{2} \, dx

5
Use Integration by Parts on ∫exx2 dx\int {e}^{x}{x}^{2} \, dx.
Let u=x2u={x}^{2}, dv=exdv={e}^{x}, du=2x dxdu=2x \, dx, v=exv={e}^{x}

6
Substitute the above into uv−∫v duuv-\int v \, du.
2ex−x2ex+∫2xex dx2{e}^{x}-{x}^{2}{e}^{x}+\int 2x{e}^{x} \, dx

7
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
2ex−x2ex+2∫xex dx2{e}^{x}-{x}^{2}{e}^{x}+2\int x{e}^{x} \, dx

8
Use Integration by Parts on ∫xex dx\int x{e}^{x} \, dx.
Let u=xu=x, dv=exdv={e}^{x}, du=dxdu=dx, v=exv={e}^{x}

9
Substitute the above into uv−∫v duuv-\int v \, du.
2ex−x2ex+2(xex−∫ex dx)2{e}^{x}-{x}^{2}{e}^{x}+2(x{e}^{x}-\int {e}^{x} \, dx)

10
The integral of ex{e}^{x} is ex{e}^{x}.
−x2ex+2xex-{x}^{2}{e}^{x}+2x{e}^{x}

11
Add constant.
−x2ex+2xex+C-{x}^{2}{e}^{x}+2x{e}^{x}+C

Done
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