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Question

Evaluate the Integral xsec2xdx


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Solution

Finding the integral:

Given, xsec2xdx

Integration by parts is done when conventional methods of integration do not work on the integrand.

The formula for integration by parts is given as,

f(x)·g(x)dx=f(x)g(x)dx-f'(x)g(x)dxdx

[General tip: When picking the functions to be substituted into the formula, f(x) must be a function that is easily differentiable or goes to zero when differentiated repeatedly. Likewise, g(x) must be the function that is more easily integrated]

Let, f(x)=x and g(x)=sec2x. Thus,

xsec2xdx=xsec2xdx-ddxxsec2xdxdx=xtan(x)-tan(x)dxsec2xdx=tan(x)=xtanx-lnsec(x)+Ctanxdx=lnsecx

Hence xsec2xdx=xtanx-lnsecx+C.


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