Integral of sec x: Solving Techniques and Tricks

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In summary, the integral of sec x is ln|sec x + tan x| + C. Common techniques for solving integrals of sec x include using trigonometric identities, substitution, and integration by parts. The most commonly used trigonometric identity for solving integrals of sec x is u-substitution, where u = sec x + tan x. Other identities such as the Pythagorean identity and the double-angle identities can also be used. The most commonly used substitution for solving integrals of sec x is u = sec x + tan x, but other substitutions can also be used. Integration by parts is a method used to find the integral of a product of two functions, with sec x being the first function and ln|sec
  • #1
Hyperreality
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I know the integral of sec x is

ln|tan(x)+sec(x)|+C,

but how would you do it? I tried all the techniques and tricks I've learned and nothing came up.
 
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  • #2
Did you try multiplying by 1?

[tex]sec(x)=\displaystyle sec(x)\frac{sec(x)+tan(x)}{tan(x)+sec(x)}=\displaystyle\frac{sec^{2}(x)+sec(x)tan(x)}{tan(x)+sec(x)}[/tex]. Your integral is now of the form [tex]\displaystyle\frac{f'(x)}{f(x)}[/tex], easy to handle.
 
  • #3
Yup. shmoe's method is the easiest one to remember.
 

FAQ: Integral of sec x: Solving Techniques and Tricks

What is the integral of sec x?

The integral of sec x is ln|sec x + tan x| + C.

What are some common techniques for solving integrals of sec x?

Common techniques for solving integrals of sec x include using trigonometric identities, substitution, and integration by parts.

How do I use trigonometric identities to solve an integral of sec x?

The most commonly used trigonometric identity for solving integrals of sec x is u-substitution, where u = sec x + tan x. Other identities such as the Pythagorean identity and the double-angle identities can also be used.

Is there a specific substitution to use for integrals of sec x?

Yes, the most commonly used substitution for solving integrals of sec x is u = sec x + tan x. However, other substitutions such as u = tan x or u = 1/cos x can also be used depending on the form of the integral.

What is the integration by parts method and how can it be used to solve integrals of sec x?

Integration by parts is a method used to find the integral of a product of two functions. To use this method for integrals of sec x, you would choose sec x as the first function and use the identity ∫sec x dx = ln|sec x + tan x| + C as the second function.

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