Primitive of Reciprocal of Cosine of a x/Logarithm of Tangent Form

Theorem

$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$

Proof

\(\ds \int \frac {\d x} {\cos x}\)

\(=\)

\(\ds \int \sec x \rd x\)

Definition of Real Secant Function

\(\ds \)

\(=\)

\(\ds \ln \size {\map \tan {\frac \pi 4 + \frac x 2} } + C\)

Primitive of $\sec x$: Tangent plus Angle Form

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {\cos a x}\)

\(=\)

\(\ds \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\)

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\sin a x}$: $\tan$ form

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.375$