Primitive of Square of Sine of a x over Cosine of a x

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Theorem

$\ds \int \frac {\sin^2 a x \rd x} {\cos a x} = \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$


Proof

\(\ds \int \frac {\sin^2 a x \rd x} {\cos a x}\) \(=\) \(\ds \int \frac {\paren {1 - \cos^2 a x} \rd x} {\cos a x}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \int \frac {\d x} {\cos a x} - \int \cos a x \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \int \sec a x \rd x - \int \cos a x \rd x\) Secant is Reciprocal of Cosine
\(\ds \) \(=\) \(\ds \frac {-\sin a x} a + \int \sec a x \rd x + C\) Primitive of $\cos a x$
\(\ds \) \(=\) \(\ds \frac {-\sin a x} a + \frac 1 a \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\) Primitive of $\sec a x$

$\blacksquare$


Also see


Sources