Integral of \( \displaystyle \sin^{2}{\left(x + 1 \right)} \)

Problem 4.219 · easy

1. \[ \int \sin^{2}{\left(x + 1 \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(\frac{1}{2} - \frac{\cos{\left(2 x + 2 \right)}}{2}\right)\, dx \]
   trig-identityUse the power-reduction identity sin(u)^2 = (1 - cos(2u))/2.✓ Proved
3. \[ = \int \frac{1}{2}\, dx - \int \frac{\cos{\left(2 x + 2 \right)}}{2}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = \frac{x}{2} - \int \frac{\cos{\left(2 x + 2 \right)}}{2}\, dx \]
   antiderivativeIntegrate the first term.✓ Proved
5. \[ = \frac{x}{2} - \frac{\sin{\left(2 x + 2 \right)}}{4} \]
   antiderivativeIntegrate the second term using the substitution rule for cos(ax+b).✓ Proved