How to solve ∫ sec^3x dx ?

Question

Charles Arocha · Accepted Answer

#int sec^3xdx=1/2(secxtanx + ln|secx+tanx|)+C# Let's solve #int secxdx#: #t=secx+tanx => dt=(secxtanx+sec^2x)dx# #dt=secx(secx+tanx)dx# #int secxdx = int secx (secx+tanx)/(secx+tanx) dx=int dt/t# #int secxdx = ln|secx+tanx| +C# #I = int sec^3xdx = int sec^2x secxdx# #u=secx => du=secxtanxdx# #dv=sec^2xdx => v=int sec^2xdx=tanx# #int sec^3xdx=secxtanx - int tanx secx tanx dx# #int sec^3xdx=secxtanx - int tan^2x secx dx# #int sec^3xdx=secxtanx - int (sec^2x-1) secx dx# #int sec^3xdx=secxtanx - int (sec^3x-secx) dx# #int sec^3xdx=secxtanx - int sec^3x dx + int secx dx# #int sec^3xdx + int sec^3x dx=secxtanx + ln|secx+tanx|# #2 int sec^3xdx=secxtanx + ln|secx+tanx|# #int sec^3xdx=1/2(secxtanx + ln|secx+tanx|)+C#

William Abdalla · Answer

undefined To solve ∫ sec^3x dx, use integration by parts. Let u = sec(x) and dv = sec^2(x) dx. Then, differentiate u to find du and integrate dv to find v. After finding du and v, apply the integration by parts formula:

∫ u dv = uv - ∫ v du

Substitute the values of u, dv, du, and v into the formula and solve for the integral. This will give you the solution to ∫ sec^3x dx.