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

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

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

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

arctan (sec (x)) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ^{2} + 1} d u

= \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= arctan (u)

Substitute back

u = sec (x)

= arctan (sec (x))

Add a constant to the solution

= arctan (sec (x)) + C

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