Integral of \( \cos^2 x \)

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Find the integral \[ \int \cos^2 x \; dx \] Use the trigonometric identity \( \; \cos^2 x = \dfrac{1}{2} (1 + \cos (2x)) \) to write
\[ \int \cos^2 x \; dx = \dfrac{1}{2} \int (1 + \cos (2x)) \; dx\]
Use the sum
rule of integrals \( \quad \displaystyle \int (f(x) + g(x) ) dx = \int f(x) dx + \int g(x) dx \) to rewrite the integral as \[ \int \cos^2 x \; dx = \dfrac{1}{2} \int dx + \int \cos (2x)) \; dx \] Use the common integrals \( \displaystyle \int \; dx = x \) and \( \displaystyle \int \cos (2x) dx = \dfrac{1}{2} \sin (2x) \) to write the final result as \[ \boxed { \int \cos^2 x \; dx = \dfrac{1}{2} x + \dfrac{1}{4} \sin (2x) + c} \]



More References and Links

  1. University Calculus - Early Transcendental - Joel Hass, Maurice D. Weir, George B. Thomas, Jr., Christopher Heil - ISBN-13 : 978-0134995540
  2. Calculus - Gilbert Strang - MIT - ISBN-13 : 978-0961408824
  3. Calculus - Early Transcendental - James Stewart - ISBN-13: 978-0-495-01166-8