Integral: Solve $\int \frac{\sin x+\cos x}{\sec x+ \tan x}dx$

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In summary, the conversation discusses how to solve the integral \int \frac{\sin x+\cos x}{\sec x+ \tan x}dx by using trigonometric identities and substitutions. The correct identity for secant is clarified and the suggestion is made to express everything in terms of sines and cosines. Further hints are given for solving the integral, including using the substitution u=2x and the Weierstrass substitution.
  • #1
alba_ei
39
1

Homework Statement


[tex] \int \frac{\sin x+\cos x}{\sec x+ \tan x}dx [/tex]

Homework Equations



[tex] \sin x = \frac{1}{\sec x} [/tex]
[tex] \cos x = \frac{\sin x}{\tan x} [/tex]


The Attempt at a Solution


i separate and try to use identities but i got nothign

1/(secx^2+secx tan x)+sin x/tanx^2+sec x :confused:
 
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  • #2
write everything in terms of sin x and cos x only, then integrate with change of variable.
 
  • #3
One of your identities is incorrect: sec(x)=1/cos(x).

As has been said above, you should first look to express everything in terms of sines and cosines. See if this gives you a hint as to how to proceed.
 
  • #4
please tell more

this is the shape that i got
don't know how to complete

–integral sin²x-sin2x-1/2(1+sinx)
 
  • #5
[tex]-(\int \sin^2 x dx - \int \sin 2x dx - 1/2\int 1+\sin x)[/tex]

Write sin^2 x as (1/2) (1-cos2x).

for sin 2x, make a substitution u=2x, then remember the integral of sin u is -cos u.

For the 3rd one, if you can't do it, why are you doing this question?
 
  • #6
you can also try weierstrass substitution
i.e t = tan x/2
 

1. What is the process for solving this integral?

The first step in solving this integral is to rewrite the expression using trigonometric identities. In this case, we can use the identity sec x = 1/cos x and tan x = sin x/cos x. This will give us sin x + cos x in the numerator and 1/cos x + sin x/cos x in the denominator. We can then simplify the expression to (sin x + cos x)/cos x. From here, we can use the substitution method to solve the integral.

2. What is the substitution method and how is it used to solve this integral?

The substitution method is a commonly used technique in solving integrals. It involves substituting a variable for a more complex expression in order to simplify the integral. In this case, we can let u = sin x + cos x, which will give us du = (cos x - sin x)dx. We can then replace sin x + cos x in the integral with u and cos x - sin x with du. This will give us a much simpler integral to solve.

3. What are the steps for using the substitution method to solve this integral?

The steps for using the substitution method to solve this integral are as follows:

  1. Identify a complex expression within the integral that can be simplified by substitution.
  2. Choose a variable to substitute for the complex expression.
  3. Differentiate the chosen variable to find du.
  4. Replace the complex expression and its derivative in the integral with the chosen variable and du, respectively.
  5. Simplify the resulting integral and solve using basic integration techniques.

4. Can this integral be solved without using the substitution method?

Yes, this integral can also be solved using the partial fraction method. This involves breaking down the fraction into simpler parts and integrating each part separately. However, in this case, the substitution method tends to be a quicker and more efficient method for solving the integral.

5. Are there any specific limitations or restrictions when solving this integral?

Yes, when using the substitution method, it is important to check for any restrictions on the variable that was substituted. In this case, the variable u cannot equal 0, as this would result in a division by 0. Therefore, the solution to the integral should also include any necessary restrictions on the variable.

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