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

Problem 4.554 · easy

1. \[ \int \cos^{3}{\left(3 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(1 - \sin^{2}{\left(3 x \right)}\right) \cos{\left(3 x \right)}\, dx \]
   trig-identityUse the identity cos(2*theta) = 1 - 2*sin(theta)**2, specifically cos(3x)^2 = (1 + cos(6x))/2, but here we use cos(3x)^2 = 1 - sin(3x)^2 is incorrect; let's use cos(3x)^2 = (1 + cos(6x))/2 instead.✓ Proved
3. \[ = \int \frac{\left(\cos{\left(6 x \right)} + 1\right) \cos{\left(3 x \right)}}{2}\, dx \]
   trig-identityUse the power reduction identity cos(theta)^2 = (1 + cos(2*theta))/2.✓ Proved
4. \[ = \int \frac{\cos{\left(3 x \right)} \cos{\left(6 x \right)}}{2}\, dx + \int \frac{\cos{\left(3 x \right)}}{2}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
5. \[ = \int \left(\frac{\cos{\left(3 x \right)}}{4} + \frac{\cos{\left(9 x \right)}}{4}\right)\, dx + \int \frac{\cos{\left(3 x \right)}}{2}\, dx \]
   trig-identityUse the product-to-sum identity cos(A)cos(B) = 1/2(cos(A+B) + cos(A-B)).✓ Proved
6. \[ = \int \frac{\cos{\left(3 x \right)}}{4}\, dx + \int \frac{\cos{\left(3 x \right)}}{2}\, dx + \int \frac{\cos{\left(9 x \right)}}{4}\, dx \]
   linearityDistribute the 1/4 and split the integral.✓ Proved
7. \[ = \int \frac{3 \cos{\left(3 x \right)}}{4}\, dx + \int \frac{\cos{\left(9 x \right)}}{4}\, dx \]
   algebra algebraCombine the terms with cos(3*x). Simplify the coefficients.✓ Proved
8. \[ = \frac{\sin{\left(3 x \right)}}{4} + \frac{\sin{\left(9 x \right)}}{36} \]
   antiderivativeIntegrate each term separately.✓ Proved