Question

Integrate tan^4xsecxdx

Answer 1

`\displaystyle\int\tan^4x\sec x\,\mathrm dx`

`\displaystyle\int(\sec^2x-1)^2\sec x\,\mathrm dx`

`\displaystyle\int(\sec^4x-2\sec^2x+1)\sec x\,\mathrm dx`

`\displaystyle\int\sec^5x\,\mathrm dx-2\int\sec^3x\,\mathrm dx+\int\sec x\,\mathrm dx`

The power reduction formula for secant will be a great help here (unless you want to integrate by parts several times only to arrive at the same result); for
`n\\eq1`, we have


`\displaystyle\int\sec^nx\,\mathrm dx=\frac1{n-1}\sec^(n-1)x\sin x+(n-2)/(n-1)\int\sec^(n-2)x\,\mathrm dx`

which gives


`\displaystyle\frac14\sec^4x\sin x-\frac78\sec^2x\sin x+\frac18\int\sec x\,\mathrm dx`

`=\frac14\sec^4x\sin x-\frac78\sec^2x\sin x+\frac18\ln|\sec x+\tan x|+C`