What is the integral from pi/2 to pi of 4cot(x/3) ?

The integral from pi/2 to pi of 4cot(x/3) is 6ln(3)

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integral from pi/2 to pi of 4cot(x/3)

\int_{\frac{π}{2}}^{π} 4 cot (\frac{x}{3}) d x

6 ln (3)

Decimal Notation

6.59167 \dots

Solution steps

\int_{\frac{π}{2}}^{π} 4 cot (\frac{x}{3}) d x

Take the constant out:

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

= 4 \cdot \int_{\frac{π}{2}}^{π} cot (\frac{x}{3}) d x

Rewrite using trig identities

= 4 \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

= 4 \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

= 4 \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 ∣)

= 4 \cdot 3 [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{3}{2}}

Simplify

= 12 [ln ∣ u ∣]_{\frac{1}{2}}^{\frac{3}{2}}

Compute the boundaries:

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

= 12 \cdot \frac{1}{2} ln (3)

Simplify

= 6 ln (3)

n = 0 \sum \infty \frac{n}{( 2 n + 1 ) !}

f (x) = 9 x^{2} - x^{3}, at x = 2

\int e^{2} sin (x) d x

\int_{0}^{3} 3 x - 1 d x

y^{^{''}} - 9 y^{^{'}} + 2 y = x e^{x}

What is the integral from pi/2 to pi of 4cot(x/3) ?
The integral from pi/2 to pi of 4cot(x/3) is 6ln(3)