Primitive of Square of Sine of a x over Cosine of a x
Jump to navigation
Jump to search
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.408$