What is the integral of sec(ax) ?

The integral of sec(ax) is 1/a ln|tan(ax)+sec(ax)|+C

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integral of sec(ax)

\int sec (a x) d x

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

Solution steps

\int sec (a x) d x

Rewrite using trig identities

= \int \frac{1}{cos ( a x )} d x

Apply u-substitution

: \int \frac{1}{a cos ( u )} d u

= \int \frac{1}{a cos ( u )} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int \frac{1}{cos ( u )} d u

Rewrite using trig identities

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

Use the common integral:

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

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

Substitute back

u = a x

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

Add a constant to the solution

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

\frac{d}{d x} ((4 x^{2} - 14)^{- 10})

\int \frac{1}{( x ^{2} - 6 x + 25 ) ^{3}} d x

\int_{1}^{2} \frac{- 8 ln ( x )}{x ^{2}} d x

(x - 1)^{3}

f (x) = 1100 x^{2} - 0.2 x

What is the integral of sec(ax) ?
The integral of sec(ax) is 1/a ln|tan(ax)+sec(ax)|+C