\[\int \cos{x}{x}^{2} \, dx\]
1
Regroup terms.
∫x2cos⁡x dx\int {x}^{2}\cos{x} \, dx

2
Use Integration by Parts on ∫x2cos⁡x dx\int {x}^{2}\cos{x} \, dx.
Let u=x2u={x}^{2}, dv=cos⁡xdv=\cos{x}, du=2x dxdu=2x \, dx, v=sin⁡xv=\sin{x}

3
Substitute the above into uv−∫v duuv-\int v \, du.
x2sin⁡x−∫2xsin⁡x dx{x}^{2}\sin{x}-\int 2x\sin{x} \, dx

4
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
x2sin⁡x−2∫xsin⁡x dx{x}^{2}\sin{x}-2\int x\sin{x} \, dx

5
Use Integration by Parts on ∫xsin⁡x dx\int x\sin{x} \, dx.
Let u=xu=x, dv=sin⁡xdv=\sin{x}, du=dxdu=dx, v=−cos⁡xv=-\cos{x}

6
Substitute the above into uv−∫v duuv-\int v \, du.
x2sin⁡x−2(−xcos⁡x−∫−cos⁡x dx){x}^{2}\sin{x}-2(-x\cos{x}-\int -\cos{x} \, dx)

7
Use Trigonometric Integration: the integral of cos⁡x\cos{x} is sin⁡x\sin{x}.
x2sin⁡x+2xcos⁡x−2sin⁡x{x}^{2}\sin{x}+2x\cos{x}-2\sin{x}

8
Add constant.
x2sin⁡x+2xcos⁡x−2sin⁡x+C{x}^{2}\sin{x}+2x\cos{x}-2\sin{x}+C

Done
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