What is the integral of tan(3/2 x-1) ?

The integral of tan(3/2 x-1) is -2/3 ln|cos(3/2 x-1)|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of tan(3/2 x-1)

\int tan (\frac{3}{2} x - 1) d x

- \frac{2}{3} ln cos (\frac{3}{2} x - 1) + C

Solution steps

\int tan (\frac{3}{2} x - 1) d x

Apply u-substitution

= \int tan (u) \frac{2}{3} d u

Take the constant out:

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

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

Rewrite using trig identities

= \frac{2}{3} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

= \frac{2}{3} \cdot \int - \frac{1}{v} d v

Take the constant out:

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

= \frac{2}{3} (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{2}{3} (- ln ∣ v ∣)

Substitute back

= \frac{2}{3} (- ln cos (\frac{3}{2} x - 1))

Simplify

= - \frac{2}{3} ln cos (\frac{3}{2} x - 1)

Add a constant to the solution

= - \frac{2}{3} ln cos (\frac{3}{2} x - 1) + C

What is the integral of tan(3/2 x-1) ?
The integral of tan(3/2 x-1) is -2/3 ln|cos(3/2 x-1)|+C