integral of tan^5(2x)sec^3/2(2x)dx
Y^11/11 - 2/7Y^7 + 1/3Y^3
where:
Y = (1 + u^2)^(1/4)
and:
u = tan(2x)
To find the integral of tan^5(2x)sec^(3/2)(2x)dx, we can use the substitution method.
Let's set u = tan(2x). Taking the derivative of both sides with respect to x, we get du = 2sec^2(2x)dx.
Now, we need to express everything in terms of u and du. We know that sec^2(2x) = 1 + tan^2(2x), so substituting this into our expression for du, we have du = 2(1 + u^2)dx.
Rearranging the equation, we find dx = du / [2(1 + u^2)].
Next, we substitute the expressions for u and dx in terms of du into the original integral:
∫ tan^5(2x)sec^(3/2)(2x)dx = ∫ tan^5(u)sec^(3/2)(u) * (du / [2(1 + u^2)]).
Simplifying the integral, we have:
1/2 ∫ tan^5(u) * sec^(3/2)(u) * (1 / (1 + u^2)) du.
Now, we can further simplify the integral by using the identity sec^2(u) = 1 + tan^2(u). Rearranging this identity, we have 1 + tan^2(u) = sec^2(u). Taking the square root of both sides, we get √(1 + tan^2(u)) = sec(u).
Substituting this into our integral, we have:
1/2 ∫ tan^5(u) * (sec(u))^(3/2) * (1 / (1 + u^2)) du.
Now, let's substitute Y = (1 + u^2)^(1/4). Taking the derivative of both sides with respect to u, we get dY = (1/4)(1 + u^2)^(-3/4)(2u)du. Simplifying, we have dY = (1/2)(u/(1 + u^2))du.
Rearranging the equation, we find du = (2(1 + u^2)/(u))dY. Substituting this back into the integral, we have:
1/2 ∫ tan^5(u) * (sec(u))^(3/2) * [(1 / (1 + u^2)) * (2(1 + u^2)/(u))]dY.
Simplifying the expression, we have:
∫ tan^5(u) * (sec(u))^(3/2) * [(2(1 + u^2))/(u(1 + u^2))]dY.
Now, we can simplify further by canceling out the terms (1 + u^2) in the numerator and denominator:
∫ tan^5(u) * (sec(u))^(3/2) * [(2)/(u)]dY.
Finally, we can integrate this expression with respect to Y:
∫ tan^5(u) * (sec(u))^(3/2) * [(2)/(u)]dY = 2 ∫ tan^5(u) * (sec(u))^(3/2) * (1/u) dY.
This integral does not have a simple solution in terms of elementary functions. However, if you have specific numerical values for the limits of integration, you can use numerical methods or software to approximate the value of the integral.