Primitive of Reciprocal of Cube of Cosine of a x
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Theorem
- $\ds \int \frac {\d x} {\cos^3 a x} = \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C$
Proof
\(\ds \int \frac {\d x} {\cos^3 a x}\) | \(=\) | \(\ds \int \sec^3 a x \rd x\) | Secant is Reciprocal of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sec a x \tan a x} {2 a} + \frac 1 2 \int \sec a x \rd x\) | Primitive of $\sec^n a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 2 \int \sec a x \rd x\) | Secant is $\dfrac 1 \cos$ and Tangent is $\dfrac \sin \cos$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin a x} {2 a \cos^2 a x} + \frac 1 {2 a} \ln \size {\map \tan {\frac \pi 4 + \frac {a x} 2} } + C\) | 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 $\cos a x$: $14.382$