How to determine the integral `int sec x dx`
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,554 answers
starTop subjects are Math, Science, and Business
The integral `int sec x dx` has to be determined
`int sec x dx`
= `int 1/cos x dx`
= `int cos x/cos^2x dx`
= `int cos x/(1 - sin^2x) dx`
Let `sin x = y`
`dy/dx = cos x`
=> `dy = cos x*dx`
`int cos x/(1 - sin^2x) dx`
= `int 1/(1 - y^2) dy`
= `int 1/((1 - y)(1 + y)) dy`
= `int -1/((y - 1)(y + 1)) dy`
= `int (1/2)*(y - 1 - (y + 1))/((y - 1)(y + 1)) dy`
= `(1/2)int 1/(y+1) - 1/(y-1) dy`
= `(1/2)(int 1/(y+1) dy - int 1/(y-1) dy)`
= `(1/2)(ln(y + 1) - ln(y - 1))`
= `(1/2)*ln((y + 1)/(y - 1))`
substitute `y = sin x`
= `(1/2)*ln((sin x + 1)/(sin x - 1))`
The integral `int sec x dx = (1/2)*ln((sin x + 1)/(sin x - 1)) + C`
Related Questions
- Integral of sec^n(x)dx
- 1 Educator Answer
- What is the integral `int sin^6 x dx`
- 1 Educator Answer
- Determine the integral `int x^3*sqrt(x^2 - 25) dx`
- 1 Educator Answer
- Find the integral `int sec^4 x*tan^5 x dx`
- 1 Educator Answer
- Using integration by parts, we find that `int x^(n)e^(-x) dx=`
- 3 Educator Answers
calendarEducator since 2010
write12,554 answers
starTop subjects are Math, Science, and Business
The integral `int sec x dx` has to be determined.
The integral of sec x can be solved with the use of an indirect method.
Multiply sec x by `(sec x+ tan x)/(sec x + tan x)`
`int sec x dx`
= `int (sec x*(sec x+ tan x))/(sec x + tan x) dx`
= `int (sec^2 x + sec x*tan x)/(sec x + tan x) dx`
Let `sec x + tan x = y`
`dy/dx = sec x*tan x + sec^2x`
=> `dy = sec x*tan x + sec^2x dx`
This gives:
`int (sec^2 x + sec x*tan x)/(sec x + tan x) dx`
= `int 1/ y dy`
= `ln y`
Substitute `y = sec x + tan x`
= `ln(sec x + tan x)`
The integral `int sec x dx = ln|sec x + tan x| + C`
`intsec(x)dx=int(sec(x)(sec(x)+tan(x)))/(sec(x)+tan(x))dx`
LetÂ t=sec(x)+tan(x)
dt=(sec(x)tan(x)+sec^2(x))dx
Thus
`intsec(x)dx=int((sec^2(x)+sec(x)tan(x))dx)/(sec(x)+tan(x))`
`=int1/tdt`
`=ln|t|+c`
`=ln|sec(x)+tan(x)|+c`
Student Answers