What is the integral of (tan(4x))/(cos(4x)) ?

The integral of (tan(4x))/(cos(4x)) is 1/4 sec(4x)+C

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integral of (tan(4x))/(cos(4x))

\int \frac{tan ( 4 x )}{cos ( 4 x )} d x

\frac{1}{4} sec (4 x) + C

Solution steps

\int \frac{tan ( 4 x )}{cos ( 4 x )} d x

Apply u-substitution

: \int \frac{tan ( u )}{4 cos ( u )} d u

= \int \frac{tan ( u )}{4 cos ( u )} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{tan ( u )}{cos ( u )} d u

Rewrite using trig identities

= \frac{1}{4} \cdot \int tan (u) sec (u) d u

Apply u-substitution

: \int 1 d v

= \frac{1}{4} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{4} \cdot 1 \cdot v

Substitute back

= \frac{1}{4} \cdot 1 \cdot sec (4 x)

Simplify

= \frac{1}{4} sec (4 x)

Add a constant to the solution

= \frac{1}{4} sec (4 x) + C

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What is the integral of (tan(4x))/(cos(4x)) ?
The integral of (tan(4x))/(cos(4x)) is 1/4 sec(4x)+C