Integral of \( \displaystyle \left(2 x + 1\right) \cos{\left(4 x + 2 \right)} \)

Problem 4.278 · medium

1. \[ \int \left(2 x + 1\right) \cos{\left(4 x + 2 \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(2 x \cos{\left(4 x + 2 \right)} + \cos{\left(4 x + 2 \right)}\right)\, dx \]
   linearityDistribute the cosine term.✓ Proved
3. \[ = 2 \int x \cos{\left(4 x + 2 \right)}\, dx + \int \cos{\left(4 x + 2 \right)}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = \frac{x \sin{\left(4 x + 2 \right)}}{2} - 2 \int \frac{\sin{\left(4 x + 2 \right)}}{4}\, dx + \int \cos{\left(4 x + 2 \right)}\, dx \]
   parts algebraApply integration by parts to the first integral with u = x and dv = cos(4*x + 2)dx. Simplify the expression inside the parentheses.✓ Proved
5. \[ = \frac{x \sin{\left(4 x + 2 \right)}}{2} + \frac{\cos{\left(4 x + 2 \right)}}{8} + \int \cos{\left(4 x + 2 \right)}\, dx \]
   antiderivative algebraIntegrate the sine term. Simplify the signs and fractions.✓ Proved
6. \[ = \frac{x \sin{\left(4 x + 2 \right)}}{2} + \frac{\sin{\left(4 x + 2 \right)}}{4} + \frac{\cos{\left(4 x + 2 \right)}}{8} \]
   antiderivative simplifyIntegrate the remaining cosine term. Distribute the 2 and combine terms.✓ Proved