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

Problem 4.163 · medium

1. \[ \int \left(x + 1\right) \cos{\left(x + 1 \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(x \cos{\left(x + 1 \right)} + \cos{\left(x + 1 \right)}\right)\, dx \]
   algebraDistribute the term (x + 1) into the cosine.✓ Proved
3. \[ = \int x \cos{\left(x + 1 \right)}\, dx + \int \cos{\left(x + 1 \right)}\, dx \]
   linearitySplit the integral into two separate integrals.✓ Proved
4. \[ = \sin{\left(x + 1 \right)} + \int x \cos{\left(x + 1 \right)}\, dx \]
   antiderivativeIntegrate the second term cos(x + 1).✓ Proved
5. \[ = x \sin{\left(x + 1 \right)} + \sin{\left(x + 1 \right)} - \int \sin{\left(x + 1 \right)}\, dx \]
   partsApply integration by parts to the first term with u = x and dv = cos(x + 1)dx.✓ Proved
6. \[ = x \sin{\left(x + 1 \right)} + \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} \]
   antiderivative algebraIntegrate sin(x + 1). Simplify the signs.✓ Proved
7. \[ = \left(x + 1\right) \sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)} \]
   algebraFactor out sin(x + 1) to get the final form.✓ Proved