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

The integral of-3tan(x/3) is 9ln|cos(x/3)|+C

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integral of-3tan(x/3)

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

9 ln cos (\frac{x}{3}) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{3}{u} d u

= - 3 \cdot \int - \frac{3}{u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 3 (- 3 ln ∣ u ∣)

Substitute back

u = cos (\frac{x}{3})

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

Simplify

= 9 ln cos (\frac{x}{3})

Add a constant to the solution

= 9 ln cos (\frac{x}{3}) + C

\int_{3}^{7} (x - 6) x - 3 d x

\int_{1}^{e} 1

\int \frac{x ^{3}}{1 + x ^{2}} d x

x^{- \frac{1}{4}} + x^{\frac{1}{4}}

f (x) = x^{2} - 7, at x = 3

What is the integral of-3tan(x/3) ?
The integral of-3tan(x/3) is 9ln|cos(x/3)|+C