What is the integral from pi/2 to pi of 2cot(x/3) ?

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

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integral from pi/2 to pi of 2cot(x/3)

\int_{\frac{π}{2}}^{π} 2 cot (\frac{x}{3}) d x

3 ln (3)

Decimal Notation

3.29583 \dots

Solution steps

\int_{\frac{π}{2}}^{π} 2 cot (\frac{x}{3}) d x

Take the constant out:

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

= 2 \cdot \int_{\frac{π}{2}}^{π} cot (\frac{x}{3}) d x

Rewrite using trig identities

= 2 \cdot \int_{\frac{π}{2}}^{π} \frac{cos ( \frac{x}{3} )}{sin ( \frac{x}{3} )} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot 3 [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{3}{2}}

Simplify

= 6 [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{3}{2}}

Compute the boundaries:

\frac{1}{2} ln (3)

= 6 \cdot \frac{1}{2} ln (3)

Simplify

= 3 ln (3)

x \to 1 lim (\frac{x + 1}{x ^{3} + 1})

\frac{d}{d x} (0.1 x^{3} - 0.45 x^{2} + 2.43 x + 300)

\int \frac{sin ^{5} ( x )}{cos ^{4} ( x )} d x

t \to 1 + lim (t^{\frac{1}{1 - t}})

e^{(t - 2)}

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