What is the integral of (2/3 sec(x)tan(x)) ?

The integral of (2/3 sec(x)tan(x)) is 2/3 sec(x)+C

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integral of (2/3 sec(x)tan(x))

\int (\frac{2}{3} sec (x) tan (x)) d x

\frac{2}{3} sec (x) + C

Solution steps

\int \frac{2}{3} sec (x) tan (x) d x

Take the constant out:

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

= \frac{2}{3} \cdot \int sec (x) tan (x) d x

Apply u-substitution

: \int 1 d u

= \frac{2}{3} \cdot \int 1 d u

Integral of a constant:

\int a d x = a x

= \frac{2}{3} \cdot 1 \cdot u

Substitute back

u = sec (x)

= \frac{2}{3} \cdot 1 \cdot sec (x)

Simplify

= \frac{2}{3} sec (x)

Add a constant to the solution

= \frac{2}{3} sec (x) + C

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What is the integral of (2/3 sec(x)tan(x)) ?
The integral of (2/3 sec(x)tan(x)) is 2/3 sec(x)+C