The integral ∫sec2x/(sec x + tan x)9/2 dx equals (for some arbitrary constant k)

The integral ∫sec2x/(sec x + tan x)9/2 dx equals (for some arbitrary constant k)

Solutions

Expert Solution

Let I = ∫ sec²x/(sec x + tan x)^(9/2) dx

= ∫sec x .sec x. /(sec x + tan x)^(9/2) dx

Put sec x + tan x = t

sec x tan x + sec²x dx = dt

sec x (tan x + sec x) = dt

sec – tan x = 1/t

2 sec x = t + 1/t

sec x = ½ (t + 1/t)

So I = ½ ∫[(t + 1/t)dt/t]/t^(9/2)

= ½ ∫ [t^(-9/2) + t^(-13/2)] dt

= ½ [ (-2/7)t^(-7/2) – (2/11)t^(-11/2)]+ C

= -[ (t^(-7/2)/7) + t^(-11/2)/11] + C

= -[ (sec x + tan x)^(-7/2)/7 + (sec x + tan x)^(-11/2)/11]+ C

-[ (sec x + tan x)-7/2/7 + (sec x + tan x)-11/2/11]+ C

Nikhil Thakur answered 1 year ago

Next > < Previous

Related Solutions

Evaluate the integral. (Use C for the constant of integration.) (x^2-1)/(sqrt(25+x^2)*dx Evaluate the integral. (Use C...

Evaluate the integral. (Use C for the constant of integration.) (x^2-1)/(sqrt(25+x^2)*dx Evaluate the integral. (Use C for the constant of integration.) dx/sqrt(9x^2-16)^3 Evaluate the integral. (Use C for the constant of integration.) 3/(x(x+2)(3x-1))*dx

Evaluate the integral: ∫−14 / x^2√x^2+100 dx (A) Which trig substitution is correct for this integral?...

Evaluate the integral: ∫−14 / x^2√x^2+100 dx (A) Which trig substitution is correct for this integral? x=−14sec(θ) x=100sec(θ) x=10tan(θ) x=100sin(θ) x=10sin(θ) x=100tan(θ) (B) Which integral do you obtain after substituting for x and simplifying? Note: to enter θ, type the word theta. (C) What is the value of the above integral in terms of θ? (D) What is the value of the original integral in terms of x?

Solve the following integral ∫ (x^2 + x + 2) / (x + 1)(x^2 + 1) dx....

Solve the following integral ∫ (x^2 + x + 2) / (x + 1)(x^2 + 1) dx. Use partial fraction decomposition. Show all work and steps to get to solution.

Find the indefinite integral using the substitution x = 6 tan(θ). (Use C for the constant...

Find the indefinite integral using the substitution x = 6 tan(θ). (Use C for the constant of integration.) x sqrt 36 + x2 dx

Evaluate the integral. ∫4−2. (1/(x+5)((x^2)+16)) dx

The integral ∫(2x3-1)/(x4 + x) dx is equal to (here C is the constant of integration)

Evaluate each integral using trig substitutions 1.) Integral of (3x^5dx)/(sqrt(16-x^2) 2.) Integral of (sqrt(x^2-16)dx)/x 3.) Integral...

Evaluate each integral using trig substitutions 1.) Integral of (3x^5dx)/(sqrt(16-x^2) 2.) Integral of (sqrt(x^2-16)dx)/x 3.) Integral of (6dx)/(16+16x^2)

Question