What is the integral of 1/(cos^3(x)) ?

The integral of 1/(cos^3(x)) is -xsec^3(x)cos^3(x)+x+1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|+C

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integral of 1/(cos^3(x))

integral

\frac{1}{cos ^{3} ( x )}

- x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

\int \frac{1}{cos ^{3} ( x )} d x

\frac{1}{cos ^{3} ( x )} = sec^{3} (x) (1 - cos^{3} (x)) + 1

= \int sec^{3} (x) (1 - cos^{3} (x)) + 1 d x

Rewrite using trig identities

= \int \frac{1 - cos ^{3} ( x )}{cos ^{3} ( x )} + 1 d x

Apply Integration By Parts

: x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - \int 3 x sec^{4} (x) sin (x) d x

= x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - \int 3 x sec^{4} (x) sin (x) d x

\int 3 x sec^{4} (x) sin (x) d x = x sec^{3} (x) - \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - (x sec^{3} (x) - \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

Simplify

x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - (x sec^{3} (x) - \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) : - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

\frac{x}{( 1 - x )}

e^{5} x

\frac{x}{4}

arctan (\frac{1}{x})

\frac{x ^{2}}{( 1 + x )}

What is the integral of 1/(cos^3(x)) ?
The integral of 1/(cos^3(x)) is -xsec^3(x)cos^3(x)+x+1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|+C