Primitive of Square of Cosine Function
Theorem
- $\ds \int \cos^2 x \rd x = \frac x 2 + \frac {\sin 2 x} 4 + C$
where $C$ is an arbitrary constant.
Corollary
- $\ds \int \cos^2 x \rd x = \frac {x + \sin x \cos x} 2 + C$
Proof 1
| \(\ds \int \cos^2 x \rd x\) | \(=\) | \(\ds \int \paren {\frac {1 + \cos 2 x} 2} \rd x\) | Square of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int \frac 1 2 \rd x + \int \paren {\frac {\cos 2 x} 2} \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac x 2 + C + \int \paren {\frac {\cos 2 x} 2} \rd x\) | Primitive of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac x 2 + C + \frac 1 2 \int \cos 2 x \rd x\) | Primitive of Constant Multiple of Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac x 2 + \frac 1 2 \paren {\frac {\sin 2 x} 2} + C\) | Primitive of Function of Constant Multiple and Primitive of Cosine Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac x 2 + \frac {\sin 2 x} 4 + C\) | Primitive of Function of Constant Multiple and Primitive of Cosine Function |
$\blacksquare$
Proof 2
| \(\ds I_n\) | \(=\) | \(\ds \int \cos^n x \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n-2}\) | Reduction Formula for Integral of Power of Cosine | |||||||||||
| \(\ds I_0\) | \(=\) | \(\ds \int \left({\cos x}\right)^0 \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x + C\) | Primitive of Constant | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds I_2\) | \(=\) | \(\ds \frac {\cos x \sin x} 2 + \frac x 2 + \frac C 2\) | setting $n = 2$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin 2 x} 4 + \frac x 2 + C'\) | Double Angle Formula for Sine |
$\blacksquare$
Proof 3
| \(\ds \int \cos^2 x \rd x\) | \(=\) | \(\ds \frac 1 4 \int \paren {e^{i x} + e^{-i x} }^2 \rd x\) | Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 4 \int \paren {e^{2 i x} + 2 + e^{-2 i x} } \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 4 \paren {\frac{e^{2 i x} - e^{-2 i x} } {2 i} + 2 x} + C\) | Primitive of $e^{a x}$, Primitive of Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin 2 x} 4 + \frac x 2 + C\) | Euler's Sine Identity |
$\blacksquare$
Also presented as
Some sources present this as:
- $\ds \int \cos^2 x \rd x = \frac 1 2 \paren {x + \frac {\sin 2 x} 2} + C$
Sources
- 1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.22$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.22.$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals