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

Problem 4.189 · medium

1. \[ \int \sin{\left(x \right)} \cos^{3}{\left(x \right)}\, dx \]
   integral algebraStart with the integral of the given function. Rewrite the integrand to prepare for substitution.✓ Proved
2. \[ = \int \frac{\sin{\left(2 x \right)} \cos^{2}{\left(x \right)}}{2}\, dx \]
   trig-identity linearityUse the double angle identity sin(2x) = 2sin(x)cos(x). Pull the constant factor out of the integral.✓ Proved
3. \[ = \int \left(\frac{\cos{\left(2 x \right)}}{4} + \frac{1}{4}\right) \sin{\left(2 x \right)}\, dx \]
   trig-identityUse the power reduction identity for cos(x)**2.✓ Proved
4. \[ = \int \left(\frac{\sin{\left(2 x \right)} \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{4}\right)\, dx \]
   algebraDistribute the terms inside the integral.✓ Proved
5. \[ = \int \frac{\sin{\left(2 x \right)} \cos{\left(2 x \right)}}{4}\, dx + \int \frac{\sin{\left(2 x \right)}}{4}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
6. \[ = \int \frac{\sin{\left(2 x \right)}}{4}\, dx + \int \frac{\sin{\left(4 x \right)}}{8}\, dx \]
   trig-identityUse the double angle identity again for the second term.✓ Proved
7. \[ = - \frac{\cos{\left(2 x \right)}}{8} - \frac{\cos{\left(4 x \right)}}{32} \]
   antiderivativeEvaluate the remaining integrals.✓ Proved