Integral of \( \displaystyle 4 x^{2} \sin{\left(2 x \right)} \)

Problem 4.165 · medium

1. \[ \int 4 x^{2} \sin{\left(2 x \right)}\, dx \]
   integralStart with the integral of the function.✓ Proved
2. \[ = 4 \int x^{2} \sin{\left(2 x \right)}\, dx \]
   linearityFactor out the constant 4.✓ Proved
3. \[ = - 2 x^{2} \cos{\left(2 x \right)} + 4 \int x \cos{\left(2 x \right)}\, dx \]
   parts algebraApply integration by parts with u = x**2 and dv = sin(2*x)dx. Add parentheses for clarity.✓ Proved
4. \[ = - 2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} - 4 \int \frac{\sin{\left(2 x \right)}}{2}\, dx \]
   partsApply integration by parts again to the remaining integral.✓ Proved
5. \[ = - 2 x^{2} \cos{\left(2 x \right)} + 2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \]
   antiderivative simplify simplifyEvaluate the integral of sin(2*x). Simplify the signs and fractions. Distribute the 4 to get the final antiderivative.✓ Proved