\[\int (\sin{({x}^{2})})x \, dx\]
1
Regroup terms.
∫xsin⁡(x2) dx\int x\sin{({x}^{2})} \, dx

2
Use Integration by Substitution.
Let u=x2u={x}^{2}, du=2x dxdu=2x \, dx, then x dx=12 dux \, dx=\frac{1}{2} \, du

3
Using uu and dudu above, rewrite ∫xsin⁡(x2) dx\int x\sin{({x}^{2})} \, dx.
∫sin⁡u2 du\int \frac{\sin{u}}{2} \, du

4
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
12∫sin⁡u du\frac{1}{2}\int \sin{u} \, du

5
Use Trigonometric Integration: the integral of sin⁡u\sin{u} is −cos⁡u-\cos{u}.
−cos⁡u2-\frac{\cos{u}}{2}

6
Substitute u=x2u={x}^{2} back into the original integral.
−cos⁡(x2)2-\frac{\cos{({x}^{2})}}{2}

7
Add constant.
−cos⁡(x2)2+C-\frac{\cos{({x}^{2})}}{2}+C

Done
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