# Find the integral `int secx*tan^3x dx`

justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int sec x* tan^3x dx` has to be determined.

`int sec x* tan^3x dx`

=> `int (1/cos x)*(sin^3x)/(cos^3x) dx`

=> `int (sin^3x)/(cos^4x) dx`

=> int (sin x*(1 - cos^2x))/(cos^4x) dx

let `cos x = y => dy = -sin x*dx`

=> `-int ((1 - y^2)/y^4 dy`

=> `int 1/y^2 - 1/y^4 dy`

=> `-y^-1 + y^-3/3`

=> `1/(3*y^3) - 1/y`

substitute `y = cos x`

=> `1/(3*cos^3x) - 1/cos x + C`

The integral `int sec x* tan^3x dx = 1/(3*cos^3x) - 1/cos x + C`
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