Question 886572: Can u please help me integrate { xsec^2x dx
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! I assume you mean xsec^2x dx = x*sec^2(x)*dx
let int stand for integration and use integration by parts, that is
int(uv') = uv - int(vu')
here let
u = x --> u' = 1
v' = sec^2(x) --> v = tan(x)
Therefore,
int(x*sec^2(x) dx) = x*tan(x) - int((1)tan(x) dx)
= x* tan(x) - int(tan(x) dx)
for int(tan(x) dx) = int(sin(x)/cos(x) dx)
let u = cos(x) --> du/dx = -sin(x) --> dx = -du/sin(x)
our integral becomes
int(sin(x)/cos(x) dx ) = int(sin(x)/u * -du/sin(x) )
= - int(1/u du) = -ln(u) = -ln(cos(x)) + C ; C = integration constant
Therefore,
int(x*sec^2(x) dx) = xtan(x) + ln(cos(x)) + C
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