What is the integral of (sec(x)tan(x))/(sec^2(x)+1) ?

The integral of (sec(x)tan(x))/(sec^2(x)+1) is arctan(sec(x))+C

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integral of (sec(x)tan(x))/(sec^2(x)+1)

\int \frac{sec ( x ) tan ( x )}{sec ^{2} ( x ) + 1} d x

arctan (sec (x)) + C

Solution steps

\int \frac{sec ( x ) tan ( x )}{sec ^{2} ( x ) + 1} 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 (sec(x)tan(x))/(sec^2(x)+1) ?
The integral of (sec(x)tan(x))/(sec^2(x)+1) is arctan(sec(x))+C