Primitive of Power of Secant of a x by Tangent of a x

Theorem

$\ds \int \sec^n a x \tan a x \rd x = \frac {\sec^n a x} {n a} + C$

for $n \ne 0$.

Proof

\(\ds z\)

\(=\)

\(\ds \sec a x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d z} {\d x}\)

\(=\)

\(\ds a \sec a x \tan a x\)

Derivative of Secant Function, Chain Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \sec^n a x \tan a x \rd x\)

\(=\)

\(\ds \int \frac {z^{n - 1} \rd z} a\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds \frac {z^n} {n a}\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac {\sec^n a x} {n a} + C\)

substituting for $z$

$\blacksquare$

Also see

• Primitive of $\csc^n a x \cot a x$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sec a x$: $14.454$
• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $94$.