Primitive of x by Square of Tangent of a x
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Theorem
- $\ds \int x \tan^2 a x \rd x = \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} - \frac {x^2} 2 + C$
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:
\(\ds u\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds 1\) | Primitive of Power |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \tan^2 a x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {\tan a x} a - x\) | Primitive of $\tan^2 a x$ |
Then:
\(\ds \int x \tan^2 a x \rd x\) | \(=\) | \(\ds x \paren {\frac {\tan a x} a - x} - \int \paren {\frac {\tan a x} a - x} \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \tan a x} a - x^2 - \int \paren {\frac {\tan a x} a - x} \rd x + C\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \tan a x} a - x^2 - \frac 1 a \int \tan a x \rd x + \int x \rd x + C\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \tan a x} a - x^2 - \frac 1 a \paren {\frac {-\ln \size {\cos a x} } a} + \int x \rd x + C\) | Primitive of $\tan a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \tan a x} a - x^2 + \frac 1 {a^2} \ln \size {\cos a x} + \frac {x^2} 2 + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x \tan a x} a + \frac 1 {a^2} \ln \size {\cos a x} - \frac {x^2} 2 + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $x \sin^2 a x$
- Primitive of $x \cos^2 a x$
- Primitive of $x \cot^2 a x$
- Primitive of $x \sec^2 a x$
- Primitive of $x \csc^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\tan a x$: $14.437$