Integrating sec x dx: Multiply by \frac{tan x + sec x}{tan x + sec x}

In summary, to find the integral of sec x dx, you can multiply the integrand by \frac{tan x + sec x}{tan x + sec x} to get \frac{tan x.sec x + sec^2 x}{tan x + sec x}dx. Then, noticing that the numerator is the derivative of the denominator, you can rewrite it as \frac{d(sec x + tan x)}{sec x + tan x}dx. Finally, you can integrate this using the formula for \frac{du}{u}, giving you the answer of log(u), or in this case, log(sec x + tan x).
  • #1
JFonseka
117
0

Homework Statement


By multiplying the integrand sec x dx by [tex]\frac{tan x + sec x}{tan x + sec x}[/tex] find the integral of sec x dx


Homework Equations



d/dx sec x = tan x.sec x
d/dx tan x = sec^2 x

The Attempt at a Solution



sec x dx([tex]\frac{tan x + sec x}{tan x + sec x}[/tex]) =>

[tex]\frac{tan x.sec x + sec^2 x}{tan x + sec x}[/tex]dx

Just noticed the numerator is the derivative of the denominator, so =>


[tex]\frac{d(sec x + tan x)}{sec x + tan x}[/tex]dx

Not sure what to do from here...
 
Last edited:
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  • #2
That's integral of du/u where u=sec(x)+tan(x). What's integral of du/u?
 
  • #3
Well...integrating the derivative would just return the original function wouldn't it? But in this case it's the reciprocal so it would be 1/sec x + tan x ?
 
  • #4
Nooo. Integral of du/u is log(u), isn't it?
 
  • #5
Oh yeaaaaa...I got confused. I always think of it as 1/x, not dx/x.
Thanks Richard!
 

FAQ: Integrating sec x dx: Multiply by \frac{tan x + sec x}{tan x + sec x}

What does it mean to "integrate sec x dx"?

When we talk about integrating sec x dx, we are referring to finding the indefinite integral of the secant function. This means finding a function whose derivative is equal to sec x. It is a fundamental concept in calculus that allows us to solve a variety of problems involving the secant function.

Why do we need to multiply by \frac{tan x + sec x}{tan x + sec x} when integrating sec x dx?

The reason we need to multiply by \frac{tan x + sec x}{tan x + sec x} is because it is the inverse of the derivative of sec x. This is known as the "integration by substitution" method, and it helps us simplify the integral and make it easier to solve.

How do we integrate sec x dx using \frac{tan x + sec x}{tan x + sec x}?

To integrate sec x dx using \frac{tan x + sec x}{tan x + sec x}, we first rewrite the integral as \int sec x (\frac{tan x + sec x}{tan x + sec x}) dx. Then, we use the substitution u = tan x + sec x, which simplifies the integral to \int u du. From here, we can easily integrate to get the final answer.

Can we use other methods to integrate sec x dx?

Yes, there are other methods that can be used to integrate sec x dx, such as integration by parts or trigonometric substitution. However, multiplying by \frac{tan x + sec x}{tan x + sec x} is often the simplest and most efficient method for this particular integral.

What are some applications of integrating sec x dx?

Integrating sec x dx has many real-life applications, such as calculating the arc length of a curve, finding the area under a curve, and solving problems involving trigonometric functions in physics and engineering. It is an essential tool in many fields of science and mathematics.

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