Integral of \( \displaystyle 3 x \cos{\left(6 x \right)} \)

Problem 4.150 · medium

1. \[ \int 3 x \cos{\left(6 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = 3 \int x \cos{\left(6 x \right)}\, dx \]
   linearityFactor out the constant 3.✓ Proved
3. \[ = \frac{x \sin{\left(6 x \right)}}{2} - 3 \int \frac{\sin{\left(6 x \right)}}{6}\, dx \]
   parts algebraApply integration by parts with u = x and dv = cos(6*x)dx. Distribute the 3 into the parentheses.✓ Proved
4. \[ = \frac{x \sin{\left(6 x \right)}}{2} + \frac{\cos{\left(6 x \right)}}{12} \]
   antiderivative algebra simplifyEvaluate the integral of sin(6*x)/6. Simplify the signs and fractions. Distribute the 3 and simplify the final expression.✓ Proved