Use a u-substitution to rewrite the indefinite integral in terms of u and then evaluate. Give exact values only, no decimals. ∫cos(tan4x)sec^(2)4xdx=∫g(u)du.

Question

u=?

du=? dx

g(u)=?

∫g(u)du=? +C

∫cos(tan4x)sec^(2)4xdx=? +C

Answer 1

Given: Integral of cos[tan(4x)] * [sec(4x)]^2 dx

Use U-sub to solve

Solution:

u = tan(4x)

du/dx = [sec(4x)]^2 [4]

du = 4*[sec(4x)]^2 dx

I = Integral

I = Integral of cos[tan(4x)] * [sec(4x)]^2 dx

I = (1/4) * Integral of cos[u] du

Note: The (1/4) adjusts

for the 4 included in the du.

I = (1/4)*sin u + C

I = (1/4)*sin[ tan(4x) ] + C

Can't evaluate without boundaries.

Answer 2

Let u=tan4x and du becomes 4sec²4xdx. So the integral becomes ∫cos(u)du which when integrated is sin(u) +const> The integral is sin(tan4x) + const.