Primitive of Reciprocal of Square of Cosine of a x

Theorem

$\ds \int \frac {\d x} {\cos^2 a x} = \frac {\tan a x} a + C$

Corollary

$\ds \int \frac {\d x} {\cos^2 x} = \tan x + C$

Proof

\(\ds \int \frac {\d x} {\cos^2 a x}\)

\(=\)

\(\ds \int \sec^2 a x \rd x\)

Definition of Cosecant

\(\ds \)

\(=\)

\(\ds \frac {\tan a x} a + C\)

Primitive of $\sec^2 a x$

$\blacksquare$

Also see

• Primitive of $\dfrac 1 {\sin^2 a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cos a x$: $14.381$