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

Problem 4.735 · medium

1. \[ \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(x + 1\right) \cos{\left(x^{2} + 2 x + 1 \right)}\, dx \]
   algebraExpand the squared term inside the cosine.✓ Proved
3. \[ = \int \left(x \cos{\left(x^{2} + 2 x + 1 \right)} + \cos{\left(x^{2} + 2 x + 1 \right)}\right)\, dx \]
   linearityDistribute the (x + 1) term.✓ Proved
4. \[ = \int \left(x \cos{\left(\left(x + 1\right)^{2} \right)} + \cos{\left(\left(x + 1\right)^{2} \right)}\right)\, dx \]
   algebraRewrite the expanded polynomial back in its squared form for clarity.✓ Proved
5. \[ = \int x \cos{\left(\left(x + 1\right)^{2} \right)}\, dx + \int \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
   linearity algebraSplit the integral into two parts. Rewrite x as (x + 1) - 1 to facilitate substitution.✓ Proved
6. \[ = \int \left(\left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)} - \cos{\left(\left(x + 1\right)^{2} \right)}\right)\, dx + \int \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
   algebraDistribute the cosine term.✓ Proved
7. \[ = \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
   simplifyThe extra terms cancel out.✓ Proved
8. \[ = \int \frac{\left(2 x + 2\right) \cos{\left(\left(x + 1\right)^{2} \right)}}{2}\, dx \]
   rewriteExpress the integrand as a product involving the derivative of the inner function.✓ Proved
9. \[ = \int \left(x + 1\right) \cos{\left(\left(x + 1\right)^{2} \right)}\, dx \]
   simplifySimplify the coefficient.✓ Proved
10. \[ = \frac{\sin{\left(\left(x + 1\right)^{2} \right)}}{2} \]
    antiderivativeApply the substitution rule for the integral.✓ Proved