What is the integral from 0 to pi of tan(x/3) ?

The integral from 0 to pi of tan(x/3) is 3ln(2)

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integral from 0 to pi of tan(x/3)

\int_{0}^{π} tan (\frac{x}{3}) d x

3 ln (2)

Decimal Notation

2.07944 \dots

Solution steps

\int_{0}^{π} tan (\frac{x}{3}) d x

Rewrite using trig identities

= \int_{0}^{π} \frac{sin ( \frac{x}{3} )}{cos ( \frac{x}{3} )} d x

Apply u-substitution

: \int_{1}^{\frac{1}{2}} - \frac{3}{u} d u

= \int_{1}^{\frac{1}{2}} - \frac{3}{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{2}}^{1} - \frac{3}{u} d u

Take the constant out:

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

= - (- 3 \cdot \int_{\frac{1}{2}}^{1} \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - (- 3 [ln ∣ u ∣]_{\frac{1}{2}}^{1})

Simplify

= 3 [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Compute the boundaries:

ln (2)

= 3 ln (2)

\int \frac{5}{25 - x ^{2}} d x

\frac{\partial}{\partial y} (x^{4} + 5 x y^{3})

\frac{x}{7} - \frac{7}{x}

\frac{2 s + 12}{s ^{2} + 2 s + 5}

(5 x^{2} + 3)^{\frac{1}{4}}

What is the integral from 0 to pi of tan(x/3) ?
The integral from 0 to pi of tan(x/3) is 3ln(2)