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

Problem 4.377 · easy

1. \[ \int \sin^{2}{\left(2 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(\frac{1}{2} - \frac{\cos{\left(4 x \right)}}{2}\right)\, dx \]
   trig-identity algebraUse the power-reduction identity sin(theta)^2 = (1 - cos(2*theta))/2. Distribute the 1/2 into the terms.✓ Proved
3. \[ = \int \frac{1}{2}\, dx - \int \frac{\cos{\left(4 x \right)}}{2}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = \frac{x}{2} - \int \frac{\cos{\left(4 x \right)}}{2}\, dx \]
   antiderivativeIntegrate the first term.✓ Proved
5. \[ = \frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8} \]
   antiderivative simplifyIntegrate the second term using the rule for cos(ax). Simplify the final expression.✓ Proved