Primitive of Reciprocal of 1 plus Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {1 + \cos a x} = \frac 1 a \tan \frac {a x} 2 + C$
Corollary
- $\ds \int \frac {\d x} {1 + \cos x} = \tan \frac x 2 + C$
Proof
\(\ds u\) | \(=\) | \(\ds \tan \frac x 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {1 + \cos x}\) | \(=\) | \(\ds \int \frac {\dfrac {2 \rd u} {1 + u^2} } {1 + \dfrac {1 - u^2} {1 + u^2} }\) | Weierstrass Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {2 \rd u} {1 + u^2 + \paren {1 - u^2} }\) | multiplying top and bottom by $1 + u^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \d u\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds u + C\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \tan \frac x 2 + C\) | substituting for $u$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\d x} {1 + \cos a x}\) | \(=\) | \(\ds \frac 1 a \tan \frac {a x} 2 + C\) | Primitive of Function of Constant Multiple |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.386$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $76$.