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

The integral of cos(x)(sec(x)tan(x)) is -ln|cos(x)|+C

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

\int cos (x) (sec (x) tan (x)) d x

- ln ∣ cos (x) ∣ + C

Solution steps

\int cos (x) sec (x) tan (x) d x

Rewrite using trig identities

= \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

: \int - \frac{1}{u} d u

= \int - \frac{1}{u} d u

Take the constant out:

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

= - \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cos (x)

= - ln ∣ cos (x) ∣

Add a constant to the solution

= - ln ∣ cos (x) ∣ + C

x \to - 2 lim (\frac{x + 2}{∣ x + 2 ∣})

x \to 0 lim (ln (x) sin (x))

\frac{d}{d x} (ln (x - 1))

\int_{- 2}^{7} \frac{d}{d t} ((t - 7) (1 - t)) d t

\int_{0}^{0.6} 500 x d x

What is the integral of cos(x)(sec(x)tan(x)) ?
The integral of cos(x)(sec(x)tan(x)) is -ln|cos(x)|+C