Primitive of Reciprocal of q plus p by Secant of a x

Theorem

$\ds \int \frac {\d x} {q + p \sec a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \cos a x} + C$

Proof

\(\ds \int \frac {\d x} {q + p \sec a x}\)

\(=\)

\(\ds \frac 1 q \int \frac {q \rd x} {q + p \sec a x}\)

multiplying top and bottom by $q$

\(\ds \)

\(=\)

\(\ds \frac 1 q \int \frac {\paren {q + p \sec a x - p \sec a x} \rd x} {q + p \sec a x}\)

\(\ds \)

\(=\)

\(\ds \frac 1 q \int \frac {\paren {q + p \sec a x} \rd x} {q + p \sec a x} - \frac p q \int \frac {\sec a x \rd x} {q + p \sec a x}\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 q \int \d x - \frac p q \int \frac {\sec a x \rd x} {q + p \sec a x}\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac x q - \frac p q \int \frac {\sec a x \rd x} {q + p \sec a x} + C\)

Primitive of Constant

\(\ds \)

\(=\)

\(\ds \frac x q - \frac p q \int \frac {\cos a x \sec a x \rd x} {q \cos a x + p \cos a x \sec a x} + C\)

multiplying top and bottom by $\cos a x$

\(\ds \)

\(=\)

\(\ds \frac x q - \frac p q \int \frac {\d x} {p + q \cos a x} + C\)

Secant is Reciprocal of Cosine

$\blacksquare$

Also see

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

• Primitive of $\dfrac 1 {p + q \cos a x}$

• Primitive of $\dfrac 1 {p + q \tan a x}$

• Primitive of $\dfrac 1 {p + q \cot a x}$

• Primitive of $\dfrac 1 {q + p \csc a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sec a x$: $14.459$