Primitive of Reciprocal of 1 plus Cosine of a x

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

• Primitive of $\dfrac 1 {1 - \cos a x}$

• Primitive of $\dfrac 1 {1 + \sin a x}$
• Primitive of $\dfrac 1 {1 - \sin a x}$

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: $76$.