Integral of \( \displaystyle x^{2} \sin{\left(x \right)} \)

Problem 4.359 · easy

1. \[ \int x^{2} \sin{\left(x \right)}\, dx \]
   integralStart with the integral of the integrand.✓ Proved
2. \[ = - x^{2} \cos{\left(x \right)} - \int \left(- 2 x \cos{\left(x \right)}\right)\, dx \]
   partsApply integration by parts with u = x**2 and dv = sin(x) dx.✓ Proved
3. \[ = - x^{2} \cos{\left(x \right)} + \int 2 x \cos{\left(x \right)}\, dx \]
   algebraSimplify the expression by distributing the negative sign.✓ Proved
4. \[ = - x^{2} \cos{\left(x \right)} + 2 \int x \cos{\left(x \right)}\, dx \]
   linearityPull the constant 2 out of the integral.✓ Proved
5. \[ = - x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} - 2 \int \sin{\left(x \right)}\, dx \]
   parts algebraApply integration by parts again to the remaining integral with u = x and dv = cos(x) dx. Distribute the constant 2 into the parentheses.✓ Proved
6. \[ = - x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)} \]
   antiderivative algebraEvaluate the integral of sin(x). Simplify the final expression.✓ Proved