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

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

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integral of tan(2-3x)

integral

tan (2 - 3 x)

\frac{1}{3} ln ∣ cos (2 - 3 x) ∣ + C

Solution steps

\int tan (2 - 3 x) d x

Apply u-substitution

: \int - \frac{1}{3} tan (u) d u

= \int - \frac{1}{3} tan (u) d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int tan (u) d u

Rewrite using trig identities

= - \frac{1}{3} \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{3} (- ln ∣ v ∣)

Substitute back

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

Simplify

= \frac{1}{3} ln ∣ cos (2 - 3 x) ∣

Add a constant to the solution

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

e^{x^{2}} x^{2}

3 x^{2} - 9 x + 5

\frac{e ^{x}}{3}

\frac{1}{5 x}

e^{x} + \frac{1}{2} + 2 x^{2} + 3

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