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

Theorem

$\ds \int \frac {\d x} {\cos a x} = \frac 1 a \ln \size {\sec a x + \tan a z}$

Proof

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

\(=\)

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

Definition of Real Secant Function

\(\ds \)

\(=\)

\(\ds \ln \size {\sec x + \tan x} + C\)

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

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \frac 1 a \ln \size {\sec a x + \tan a x} + C\)

Primitive of Function of Constant Multiple

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\sin a x}$: $\csc - \cot$ form

Sources

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