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Evaluate the integral\(\int {\rm{x}} {\rm{ sec x tan xdx}}\).

Short Answer

Expert verified

The integral value of the given equation is\(\int {\rm{x}} {\rm{secxtanxdx = xsecx - ln|tanx + secx| + c}}\).

Step by step solution

01

Expand the equation.

\({\rm{I = }}\int {\rm{x}} {\rm{secxtanxdx}}\)

Let,\({\rm{u = x}}\;\;\;{\rm{,dv = secxtanxdx}}\)

Then,\(\;\;\;{\rm{du = dx}}\;\;\;{\rm{,v = secx}}\)

Known value \(\int {\rm{u}} {\rm{dv = u}}{\rm{.v - }}\int {\rm{v}} {\rm{du}}\)

02

Evaluate the equation.

Parts integration yields

\({\rm{I = xsecx - }}\int {{\rm{sec}}} {\rm{xdx}}\)

Known value\(\int {{\rm{secxdx = ln|tanx + secx|}}} \)

\({\rm{ = xsecx - ln|tanx + secx| + c}}\)

Therefore, the integral value of the given equation is\(\int {\rm{x}} {\rm{secxtanxdx = xsecx - ln|tanx + secx| + c}}\).

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