Calculus 2 Evaluate the integral (7.5 4)

Question

evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.)

x sec(x) tan(x) dx. please show detailed solution and box the final solution, thank you.

Answer 1

I.B.P

u = x and dv = secx tanx dx

dU = dx and V = secx

the integral becomes: x secx - integral [ secx dx ]

= x sec x - ln | secx + tanx | + C

checks by differentiation:

x secx tanx +secx - (secxtanx + sec^2)/(secx+ tanx)=

x secx tanx + secx - (secx) (tanx+secx)/(secx+tanx)=

x secx tanx + secx - secx

x secx tanx

yes it checks

the anti-derivative is x sec x - ln | secx + tanx | + C

Answer 2

integral x secx tanx dx

We have the formula integral udv = uv - integral vdu

Let u = x, dv = secx tanx dx

Then du = dx

and integral dv = integral secx tanx dx

v = integral secx tanx dx

v = secx

Then we plug in the formula

integral of x secx tanx = x secx - integral of secx

= x secx - ln(secx + tanx) + C