Question

Solve: \(\displaystyle\int \cos^2 x\, dx\)

Hint:

Use cos²x=(1+cos2x)/2, then integrate term by term.

Answer 1

Step 1: Key Formula or Approach:
There is no direct antiderivative for \(\cos^2x\) as it stands, so first rewrite it using the double angle identity \(\cos2x = 2\cos^2x-1\), which gives \(\cos^2x = \dfrac{1+\cos2x}{2}\).

Step 2: Rewriting the integral:
\[ \int\cos^2x\,dx = \int\frac{1+\cos2x}{2}\,dx = \frac12\int1\,dx + \frac12\int\cos2x\,dx \]

Step 3: Integrating each piece:
\(\displaystyle\int1\,dx = x\), and \(\displaystyle\int\cos2x\,dx = \dfrac{\sin2x}{2}\) (dividing by the derivative of the inner function \(2x\)).

Final Answer:
\[ \int\cos^2x\,dx = \frac{x}{2} + \frac{\sin2x}{4} + C \] \[ \boxed{\dfrac{x}{2}+\dfrac{\sin2x}{4}+C} \]