What is the integral of \sec^3(x) ?

jamessinatraaa

jamessinatraaa

Answered question

2021-12-17

What is the integral of sec3(x) ?

Answer & Explanation

Deufemiak7

Deufemiak7

Beginner2021-12-18Added 34 answers

I=sec3xdx
by integration by Pats with:
u=secx and dv=sec2xdx
du=secxtanxdx and v=tanx
=secxtanxsecxtan2xdx
by tan2x=sec2x1
=secxtanx(sec3xsecx)dx
since sec3xdx=I
=secxtanxI+secxdx
By adding I and secxdx=ln|secx+tanx|+C1
2I=secxtanx+ln|secx+tanx|+C1
By dividing by 2.
I=12secxtanx+12ln|secx+tamx|+C12
Hence
sec3dx=12secxtanx+12ln|secx+tanx|+C
Bob Huerta

Bob Huerta

Beginner2021-12-19Added 41 answers

1 Use Integration by Parts on sec3xdx
2 Substitute the above into uvvdu
secxtanxtan2xsecxdx
3 Use Pythagorean Identities: tan2x=sec2x1
secxtanx(sec2x1)secxdx
4 Expand.
secxtanxsec3xsecxdx
5 Use Sum Rule: f(x)+g(x)dx=f(x)dx+g(x)dx
secxtanxsec3xdx+secxdx
6 Set it as equal to the original integral sec3xdx
sec3xdx=secxtanxsec3xdx+secxdx
7 Add sec3xdx to both sides.
sec3xdx+sec3xdx=secxtanx+secxdx
8 Simplify sec3xdx+sec3xdx to 2sec3xdx
2sec3xdx=secxtanx+secxdx
9 Divide both sides by 2.
sec3xdx=secxtanx+secxdx2
10 Original integral solved.
sec

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