How to integrate sec^3x

1 Use Integration by Parts on / int / sec^{3}x /, dx∫sec

3

xdx.

Let u=/sec{x}u=secx, dv=/sec^{2}xdv=sec

2

x, du=/sec{x}/tan{x} /, dxdu=secxtanxdx, v=/tan{x}v=tanx

2 Substitute the above into uv-/int v /, duuv-∫vdu.

/sec{x}/tan{x}-/int / tan^{2}x/sec{x} /, dxsecxtanx-∫tan

2

xsecxdx

3 Use Pythagorean Identities: / tan^{2}x=/sec^{2}x-1tan

2

x=sec

2

x-1.

/sec{x}/tan{x}-/int (/sec^{2}x-1) / sec{x} /, dxsecxtanx-∫ (sec

2

x-1) secxdx

4 Expand (/sec^{2}x-1) / sec{x} (sec

2

x-1) secx.

/sec{x}/tan{x}-/int / sec^{3}x-/sec{x} /, dxsecxtanx-∫sec

3

x-secxdx

5 Use Sum Rule: / int f (x) + g (x) /, dx=/int f (x) /, dx+/int g (x) /, dx∫f (x) + g (x) dx=∫f (x) dx+∫g (x) dx.

/sec{x}/tan{x}-/int / sec^{3}x /, dx+/int / sec{x} /, dxsecxtanx-∫sec

3

xdx+∫secxdx

6 Set it as equal to the original integral / int / sec^{3}x /, dx∫sec

3

xdx.

/int / sec^{3}x /, dx=/sec{x}/tan{x}-/int / sec^{3}x /, dx+/int / sec{x} /, dx∫sec

3

xdx=secxtanx-∫sec

3

xdx+∫secxdx

7 Add / int / sec^{3}x /, dx∫sec

3

xdx to both sides.

/int / sec^{3}x /, dx+/int / sec^{3}x /, dx=/sec{x}/tan{x}+/int / sec{x} /, dx∫sec

3

xdx+∫sec

3

xdx=secxtanx+∫secxdx

8 Simplify / int / sec^{3}x /, dx+/int / sec^{3}x /, dx∫sec

3

xdx+∫sec

3

xdx to 2/int / sec^{3}x /, dx2∫sec

3

xdx.

2/int / sec^{3}x /, dx=/sec{x}/tan{x}+/int / sec{x} /, dx2∫sec

3

xdx=secxtanx+∫secxdx

9 Divide both sides by 22.

/int / sec^{3}x /, dx=/frac{/sec{x}/tan{x}+/int / sec{x} /, dx}{2}∫sec

3

xdx=

2



secxtanx+∫secxdx



10 Original integral solved.

/frac{/sec{x}/tan{x}+/int / sec{x} /, dx}{2}

2



secxtanx+∫secxdx



11 Use Trigonometric Integration: the integral of / sec{x}secx is / ln{ (/sec{x}+/tan{x}) }ln (secx+tanx).

/frac{/sec{x}/tan{x}+/ln{ (/sec{x}+/tan{x}) }}{2}

2



secxtanx+ln (secx+tanx)



12 Add constant.

/frac{/sec{x}/tan{x}+/ln{ (/sec{x}+/tan{x}) }}{2}+C

2



secxtanx+ln (secx+tanx)

+ C