Primitive of Reciprocal of Sine of a x plus Cosine of a x/Examples/sin x + cos x

Example of Use of Primitive of $\dfrac 1 {\sin a x + \cos a x}$

$\ds \int \dfrac {\d x} {\sin x + \cos x} = \frac 1 {\sqrt 2} \ln \size {\map \cosec {x + \frac \pi 4} - \map \cot {x + \frac \pi 4} } + C$

Proof

From Primitive of $\dfrac 1 {\sin a x + \cos a x}$:

$\ds \int \frac {\d x} {\sin a x + \cos a x} = \frac 1 {a \sqrt 2} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 8} } + C$

The result follows on setting $a = 1$.

\(\ds \int \dfrac {\d x} {\sin x + \cos x}\)

\(=\)

\(\ds \frac 1 {\sqrt 2} \ln \size {\map \tan {\frac x 2 + \frac \pi 8} } + C\)

\(\ds \)

\(=\)

\(\ds \frac 1 {\sqrt 2} \ln \size {\map \cosec {x + \frac \pi 4} - \map \cot {x + \frac \pi 4} } + C\)

after algebra

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $27$.