Please show working clearly on how you use the first part to answer the second partThanks. Created by tan. science-mathematics-en - mathematics-en

Evaluate d/dx(tan^3x) Hence evaluate ∫sec^4x dx?

May 2021 2 164 Report

Evaluate d/dx(tan^3x) Hence evaluate ∫sec^4x dx?

Please show working clearly on how you use the first part to answer the second part

Thanks

d/dx tan(x)^3 = 3 * tan(x)^2 * sec(x)^2 => 3 * (sec(x)^2 - 1) * sec(x)^2 = 3 * sec(x)^4 - 3 * sec(x)^2

int(sec(x)^4 * dx) =>

int((sec(x)^4 - sec(x)^2 + sec(x)^2) * dx) =>

int((sec(x)^4 - sec(x)^2) * dx) + int(sec(x)^2 * dx) =>

(1/3) * 3 * int((sec(x)^4 - sec(x)^2) * dx) + int(sec(x)^2 * dx) =>

(1/3) * int((3 * sec(x)^4 - 3 * sec(x)^2) * dx) + int(sec(x)^2 * dx) =>

(1/3) * tan(x)^3 + int(sec(x)^2 * dx) =>

(1/3) * tan(x)^3 + tan(x) + C

d/dx (tan³x) = 3 tan²x * sec²x = 3 (sec²x − 1) * sec²x = 3 (sec⁴x − sec²x)

∫ sec⁴x dx

= ∫ (sec⁴x − sec²x) dx + ∫ sec²x dx

= 1/3 ∫ 3 (sec⁴x − sec²x) dx + ∫ sec²x dx

= 1/3 tan³x + tanx + C