¿Cómo integro [math] \ displaystyle \ int \ dfrac {dx} {\ sin x + \ sec x} [/ math]?

Answer 1

A2A

[matemáticas] \ displaystyle {\ begin {align} \ int \ frac {1} {\ sec x + \ sin x} \, dx & = \ int \ dfrac {2 \ cos x} {2 + 2 \ sin x \ cos x } \, dx \\ & = \ int \ left (\ dfrac {\ cos x + \ sin x} {3 – (\ sin x- \ cos x) ^ 2} + \ dfrac {\ cos x- \ sin x} {1 + (\ sen x + \ cos x) ^ 2} \ right) \, dx \\ & = \ int \ frac {du} {3-u ^ 2} + \ int \ frac {dz} {1 + z ^ 2} \ qquad \ left [\ porque u = \ sin x- \ cos x \ text {&} z = \ sin x + \ cos x \ right] \\ & = \ left (\ frac {1} {2 \ sqrt {3}} \ right) \ log \ left | \ frac {\ sqrt {3} + u} {\ sqrt {3} -u} \ right | + \ tan ^ {- 1} (z) \\ & = \ left (\ frac {1} {2 \ sqrt {3}} \ right) \ log \ left | \ frac {\ sqrt {3} + (\ sin x- \ cos x)} {\ sqrt {3} – (\ sin x- \ cos x)} \ right | + \ tan ^ {- 1} \ left (\ sqrt {2} \ sin \ left (x + \ frac {\ pi} {4} \ right) \ right ) + \ text {C} \ end {align}} [/ math]

[math] \ underline {\ hspace {8 in}} [/ math]

[matemáticas] \ displaystyle \ implica \ boxed {\ int \ frac {1} {\ sec x + \ sin x} \, dx = \ left (\ frac {1} {2 \ sqrt {3}} \ right) \ log \ left | \ frac {\ sqrt {3} + (\ sin x- \ cos x)} {\ sqrt {3} – (\ sin x- \ cos x)} \ right | + \ left [\ tan ^ { -1} \ left (\ sqrt {2} \ sin \ left (x + \ frac {\ pi} {4} \ right) \ right) \ right] + \ text {C}} \ tag * {} [/ math ]

SALUDOS !, espero que mi respuesta sea correcta si no, entonces comente a continuación los errores.

[matemáticas] \ enorme \ ddot \ sonrisa [/ matemáticas]

VM