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

The integral from 0 to pi of 3tan(x/3) is 9ln(2)

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

\int_{0}^{π} 3 tan (\frac{x}{3}) d x

9 ln (2)

Decimal Notation

6.23832 \dots

Solution steps

\int_{0}^{π} 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_{0}^{π} tan (\frac{x}{3}) d x

Rewrite using trig identities

= 3 \cdot \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

= 3 \cdot \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

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

Simplify

= 9 [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Compute the boundaries:

ln (2)

= 9 ln (2)

\frac{d}{d x} (1 + sec^{2} (x))

x \to 2 lim (\frac{x - 9}{x - 9 x})

\int_{4}^{7} x^{3} ln (4 x) d x

e^{\frac{7}{u}}

\frac{d x}{d t} = x - 2

What is the integral from 0 to pi of 3tan(x/3) ?
The integral from 0 to pi of 3tan(x/3) is 9ln(2)