What is the integral of 1/(sec^4(x)) ?

The integral of 1/(sec^4(x)) is cos^3(x)sin(x)+3/8 (x-1/4 sin(4x))+C

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integral of 1/(sec^4(x))

\int \frac{1}{sec ^{4} ( x )} d x

cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) + C

Solution steps

\int \frac{1}{sec ^{4} ( x )} d x

Rewrite using trig identities

= \int cos^{4} (x) d x

Simplify

cos^{4} (x) : cos^{3} (x) cos (x)

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

Apply Integration By Parts

= cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x

\int - 3 cos^{2} (x) sin^{2} (x) d x = - \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

Simplify

= cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))

Add a constant to the solution

= cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x)) + C

s l o p e (4.4) (2.4)

\int \frac{sin ( x )}{x} d x

\int cos (\frac{π}{2}) x d x

x \to 2 lim (5 - x^{3})

\frac{d}{d x} (\frac{tan ( x ) - 4}{sec ( x )})

What is the integral of 1/(sec^4(x)) ?
The integral of 1/(sec^4(x)) is cos^3(x)sin(x)+3/8 (x-1/4 sin(4x))+C