Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine

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Theorem

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


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:

\(\ds u\) \(=\) \(\ds \frac {\sin^{m - 1} a x} {\cos^n a x}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {\cos^n a x \dfrac {\d} {\d x} \sin^{m - 1} a x - \sin^{m - 1} a x \dfrac {\d} {\d x} \cos^n a x} {\cos^{2 n} a x}\) Quotient Rule for Derivatives


Thus:

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

and so:

\(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {\cos^n a x \paren {a \paren {m - 1} \sin^{m - 2} a x \cos a x} + \sin^{m - 1} a x \paren {a n \cos^{n - 1} a x \sin a x} } {\cos^{2 n} a x}\)
\(\ds \) \(=\) \(\ds \frac {a \paren {m - 1} \sin^{m - 2} a x \cos^2 a x + a n \sin^m a x} {\cos^{n + 1} a x}\) simplifying and cancelling $\cos^{n - 1}$
\(\ds \) \(=\) \(\ds \frac {a \sin^{m - 2} a x \paren {\paren {m - 1} \cos^2 a x + n \sin^2 a x} } {\cos^{n + 1} a x}\) factorising
\(\ds \) \(=\) \(\ds \frac {a \sin^{m - 2} a x \paren {\paren {m - 1} \paren {1 - \sin^2 a x} + n \sin^2 a x} } {\cos^{n + 1} a x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \frac {a \sin^{m - 2} a x \paren {\paren {n - m + 1} \sin^2 a x + \paren {m - 1} } } {\cos^{n + 1} a x}\) simplifying
\(\ds \) \(=\) \(\ds \frac {a \paren {n - m + 1} \sin^m a x} {\cos^{n + 1} a x} + \frac {a \paren {m - 1} \sin^{m - 2} a x} {\cos^{n + 1} a x}\) separating


Then let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \sin a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {-\cos a x} a\) Primitive of $\sin a x$


Then:

\(\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x\) \(=\) \(\ds \int \frac {\sin^{m - 1} a x} {\cos^n a x} \sin a x \rd x\)
\(\ds \) \(=\) \(\ds \paren {\frac {\sin^{m - 1} a x} {\cos^n a x} } \paren {\frac {-\cos a x} a}\) Integration by Parts
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \int \paren {\frac {-\cos a x} a} \paren {\frac {a \paren {n - m + 1} \sin^m a x} {\cos^{n + 1} a x} + \frac {a \paren {m - 1} \sin^{m - 2} a x} {\cos^{n + 1} a x} } \rd x + C\)
\(\ds \) \(=\) \(\ds \frac {-\sin^{m - 1} a x} {a \cos^{n - 1} a x}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {n - m + 1} \int \frac {\sin^m a x} {\cos^n a x} \rd x\) Linear Combination of Primitives
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \paren {m - 1} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C\)
\(\ds \leadsto \ \ \) \(\ds \paren {1 - \paren {n - m + 1} } \int \frac {\sin^m a x} {\cos^n a x} \rd x\) \(=\) \(\ds \frac {-\sin^{m - 1} a x} {a \cos^{n - 1} a x} + \paren {m - 1} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C\)
\(\ds \leadsto \ \ \) \(\ds \paren {m - n} \int \frac {\sin^m a x} {\cos^n a x} \rd x\) \(=\) \(\ds \frac {-\sin^{m - 1} a x} {a \cos^{n - 1} a x} + \paren {m - 1} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C\)
\(\ds \leadsto \ \ \) \(\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x\) \(=\) \(\ds \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C\)

$\blacksquare$


Sources

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