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Question

Integrate sec3xdx

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Solution

Consider the given integral.

I=sec3xdx

I=secxsec2xdx

We know that,

u.v=uvdx(ddx(u).vdx)dx

Therefore,

I=secxtanxsecxtanxtanxdx

I=secxtanxsecxtan2xdx

I=secxtanxsecx(sec2x1)dx

I=secxtanx(sec3xsecx)dx

I=secxtanx(sec3x)dx+secxdx

I=secxtanxI+secxdx

2I=secxtanx+log|secx+tanx|+C

2I=secxtanx+log|secx+tanx|+C

I=12[secxtanx+log|secx+tanx|]+C

Hence, the value of integral is .12[secxtanx+log|secx+tanx|]+C.


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