What is the integral of 1/(cos(5x)) ?

The integral of 1/(cos(5x)) is 1/5 ln|tan(5x)+sec(5x)|+C

Popular >

integral of 1/(cos(5x))

\int \frac{1}{cos ( 5 x )} d x

\frac{1}{5} ln ∣ tan (5 x) + sec (5 x) ∣ + C

Solution steps

\int \frac{1}{cos ( 5 x )} d x

Rewrite using trig identities

= \int sec (5 x) d x

Apply u-substitution

: \int sec (u) \frac{1}{5} d u

= \int sec (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{5} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = 5 x

= \frac{1}{5} ln ∣ tan (5 x) + sec (5 x) ∣

Add a constant to the solution

= \frac{1}{5} ln ∣ tan (5 x) + sec (5 x) ∣ + C

\int \frac{2}{4 x} d x

\int \frac{2 y + 3}{( y - 1 ) ( y ^{3} - 1 )} d y

\int t (sec^{2} (t^{2})) (tan^{4} (t^{2})) d t

\int (cos^{2} (y)) d y

\int \frac{1}{x ^{2}} - 6 x d x

What is the integral of 1/(cos(5x)) ?
The integral of 1/(cos(5x)) is 1/5 ln|tan(5x)+sec(5x)|+C