Primitive of x over Power of Cosine of a x

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Theorem

$\ds \int \frac {x \rd x} {\cos^n a x} = \frac {x \sin a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cos^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x} + C$


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \frac x {\cos^{n - 2} a x}\)
\(\ds \) \(=\) \(\ds x \cos^{- n + 2} a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds x \map {\frac \d {\d x} } {\cos^{-n + 2} a x} + \paren {\frac \d {\d x} x} \paren {\cos^{-n + 2} a x}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds -a x \paren {-n + 2} \cos^{-n + 1} a x \sin a x + \cos^{-n + 2} a x\) Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {a x \paren {n - 2} \sin a x} {\cos^{n - 1} a x} + \cos^{-n + 2} a x\) simplifying


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {\cos^2 a x}\)
\(\ds \) \(=\) \(\ds \sec^2 a x\) Secant is $\dfrac 1 \cos$
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {\tan a x} a\) Primitive of $\sec^2 a x$


Then:

\(\ds \int \frac {x \rd x} {\cos^n a x}\) \(=\) \(\ds \paren {\frac x {\cos^{n - 2} a x} } \paren {\frac {\tan a x} a}\) Integration by Parts
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \int \paren {\frac {\tan a x} a} \paren {\frac {a x \paren {n - 2} \sin a x} {\cos^{n - 1} a x} + \cos^{- n + 2} a x} \rd x\)
\(\ds \) \(=\) \(\ds \frac {x \sin a x} {a \cos^{n - 1} a x} - \int \paren {\frac {x \paren {n - 2} \sin^2 a x} {\cos^n a x} -\frac {\sin a x} {a \cos^{n - 1} a x} } \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac {x \sin a x} {a \cos^{n - 1} a x} - \paren {n - 2} \int \frac {x \sin^2 a x} {\cos^n a x} \rd x + \frac 1 a \int \frac {\sin a x}{\cos^{n - 1} a x} \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {x \sin a x} {a \cos^{n - 1} a x}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {n - 2} \int \frac {x \paren {1 - \cos^2 a x} } {\cos^n a x} \rd x\) Sum of Squares of Sine and Cosine
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac 1 a \paren {\frac {-1} {\paren {n - 2} \cos^{n - 2} a x} }\) Primitive of $\cos^n a x \sin a x$
\(\ds \) \(=\) \(\ds \frac {x \sin a x} {a \cos^{n - 1} a x}\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {n - 2} \int \frac {x \rd x} {\cos^n a x} + \paren {n - 2} \int \frac {x \rd x} {\cos^{n - 2} a x}\) Linear Combination of Primitives and simplifying
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac 1 a \paren {\frac {-1} {\paren {n - 2} a \cos^{n - 2} a x} }\)


This leads to:

\(\ds \paren {n - 1} \int \frac {x \rd x} {\cos^n a x}\) \(=\) \(\ds \frac {x \sin a x} {a \cos^{n - 1} a x} + \paren {n - 2} \int \frac {x \rd x} {\cos^{n - 2} a x} - \frac 1 {a^2 \paren {n - 2} \cos^{n - 2} a x}\)
\(\ds \leadsto \ \ \) \(\ds \int \frac {x \rd x} {\cos^n a x}\) \(=\) \(\ds \frac {x \sin a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cos^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cos^{n - 2} a x} + C\)

$\blacksquare$


Also see


Sources

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