Primitive of x over 1 plus Sine of a x

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Theorem

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


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:

\(\ds u\) \(=\) \(\ds x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds 1\) Derivative of Identity Function


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {1 + \sin a x}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2}\) Primitive of $\dfrac 1 {1 + \sin a x}$


Then:

\(\ds \int \frac {x \rd x} {1 + \sin a x}\) \(=\) \(\ds x \paren {-\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} } - \int \paren {-\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} } \times 1 \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds -\frac x a \map \tan {\frac \pi 4 - \frac {a x} 2} + \frac 1 a \int \map \tan {\frac \pi 4 - \frac {a x} 2} \rd x + C\) simplifying


Then:

\(\ds z\) \(=\) \(\ds \frac \pi 4 - \frac {a x} 2\)
\(\ds \frac {\d z} {\d x}\) \(=\) \(\ds -\frac a 2\) Derivative of Power
\(\ds \leadsto \ \ \) \(\ds \frac 1 a \int \map \tan {\frac \pi 4 - \frac {a x} 2} \rd x\) \(=\) \(\ds -\frac 1 a \int \frac 2 a \tan z \rd z\) Integration by Substitution
\(\ds \) \(=\) \(\ds -\frac {-2} {a^2} \ln \size {\cos z} + C\) Primitive of $\tan z$: Cosine Form
\(\ds \) \(=\) \(\ds \frac 2 {a^2} \ln \size {\map \cos {\frac \pi 4 - \frac {a x} 2} } + C\) substituting back for $z$
\(\ds \) \(=\) \(\ds \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 2 - \paren {\frac \pi 4 - \frac {a x} 2} } } + C\) Sine of Complement equals Cosine
\(\ds \) \(=\) \(\ds \frac 2 {a^2} \ln \size {\map \sin {\frac \pi 4 + \frac {a x} 2} } + C\)


Putting it all together:

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

$\blacksquare$


Also see


Sources

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