Evaluating the trigonometric integral &#x222B;<!-- ∫ --> cos &#x2061;<!-

Addison Trujillo 2022-07-10 Answered
Evaluating the trigonometric integral cos x 1 + cos x d x
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

gutinyalk
Answered 2022-07-11 Author has 11 answers
Hint Here's one method: Rewrite the integrand as
cos x 1 + cos x = 1 1 1 + cos x ,
so that the integral becomes
cos x 1 + cos x d x = ( 1 1 1 + cos x ) d x = x d x 1 + cos x .
Now, the remaining integral can be handled by exploiting the Pythagorean identity:
1 1 + cos x = 1 1 + cos x 1 cos x 1 cos x = 1 cos x 1 cos 2 x = 1 cos x sin 2 x .
Additional hint Now,
1 cos x sin 2 x d x = d x sin 2 x cos x sin 2 x d x .
The first integral is elementary, and the second can be handled with a straightforward substitution.
Did you like this example?
Subscribe for all access
Dayanara Terry
Answered 2022-07-12 Author has 4 answers
Hint cos ( x ) = 1 tan 2 ( x / 2 ) 1 + tan 2 ( x / 2 ) so the integral reduces to 0.5 ( 1 tan 2 ( x / 2 ) ) d x where we can then use identity tan 2 ( x / 2 ) = sec 2 ( x / 2 ) 1 to get the final integral which is x tan ( x / 2 ) + C
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions