Evaluate the integral: I = integral cos² x dx

Evaluate the integral: $$ I = \int \cos^2 x\,dx. $$

Step 1 · Apply the power-reduction identity

Recall that:

$$ \cos^2x = \frac{1+\cos2x}{2}. $$

Substituting this into the integral:

$$ I = \int \frac{1+\cos2x}{2}\,dx. $$

Step 2 · Split the integral

$$ I = \frac12\int 1\,dx + \frac12\int \cos2x\,dx. $$

Step 3 · Integrate each term

$$ \int 1\,dx = x, \qquad \int \cos2x\,dx = \frac{\sin2x}{2}. $$

Combining:

$$ I = \frac{x}{2} + \frac{\sin2x}{4} + C. $$

Summary Table

Additional Notes

• The integrand is defined for all real $x$.
• This is the mirror image of Problem 121: the only difference between $\int\sin^2x\,dx$ and $\int\cos^2x\,dx$ is the sign of the $\sin2x$ term, since the two power-reduction identities differ only by that sign.
• The result can be checked by differentiating $\tfrac{x}{2}+\tfrac{\sin2x}{4}$ and confirming it returns $\cos^2x$.