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

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

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

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

\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Rewrite using trig identities

= \int sec^{3} (x) d x

Apply Integral Reduction

= \frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} \cdot \int sec (x) d x

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

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

Simplify

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

= \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

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

y = 1 - x^{2}, y = 11 - 7 x

\int_{0}^{\frac{π}{2}} - 4 sin^{2} (3 x) d x

\int \frac{1}{x ^{2}} d x

x \to 0 lim (\frac{tan ^{2} ( x )}{1 - cos ( x )})

\frac{x ^{2}}{x + 2}

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