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

Problem 4.358 · easy

1. \[ \int \cos^{3}{\left(x \right)}\, dx \]
   integral algebraStart with the integral of the function. Rewrite the power as a product.✓ Proved
2. \[ = \int \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}\, dx \]
   trig-identityUse the identity cos(x)**2 = 1 - sin(x)**2.✓ Proved
3. \[ = - \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx + \int \cos{\left(x \right)}\, dx \]
   linearityDistribute the cos(x) and split the integral.✓ Proved
4. \[ = \sin{\left(x \right)} - \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx \]
   antiderivative substitution algebraIntegrate the first term. Express the integrand in terms of the derivative of sin(x). The derivative of sin(x) is cos(x).✓ Proved
5. \[ = - \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)} \]
   antiderivativeIntegrate the remaining term using the power rule for substitution.✓ Proved