What is the integral of (tan(x))/(cos(x)sec(sec(x))) ?

The integral of (tan(x))/(cos(x)sec(sec(x))) is sin(sec(x))+C

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integral of (tan(x))/(cos(x)sec(sec(x)))

\int \frac{tan ( x )}{cos ( x ) sec ( sec ( x ) )} d x

sin (sec (x)) + C

Solution steps

\int \frac{tan ( x )}{cos ( x ) sec ( sec ( x ) )} d x

Rewrite using trig identities

= \int \frac{sin ( x ) cos ( \frac{1}{c o s ( x )} )}{cos ^{2} ( x )} d x

Apply u-substitution

= \int - \frac{cos ( \frac{1}{u} )}{u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{cos ( \frac{1}{u} )}{u ^{2}} d u

Apply u-substitution

= - \int - cos (v) d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \int cos (v) d v)

Use the common integral:

\int cos (v) d v = sin (v)

= - (- sin (v))

Substitute back

= - (- sin (\frac{1}{cos ( x )}))

Simplify

- (- sin (\frac{1}{cos ( x )})) : sin (sec (x))

= sin (sec (x))

Add a constant to the solution

= sin (sec (x)) + C

What is the integral of (tan(x))/(cos(x)sec(sec(x))) ?
The integral of (tan(x))/(cos(x)sec(sec(x))) is sin(sec(x))+C