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

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

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

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

4 ln (2)

Decimal Notation

2.77258 \dots

Solution steps

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

Rewrite using trig identities

= 4 \cdot \int_{\frac{π}{2}}^{π} \frac{cos ( \frac{x}{2} )}{sin ( \frac{x}{2} )} d x

Apply u-substitution

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

= 4 \cdot \int_{\frac{2}{2}}^{1} \frac{2}{u} d u

Take the constant out:

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

= 4 \cdot 2 \cdot \int_{\frac{2}{2}}^{1} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

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

Simplify

4 \cdot 2 [ln ∣ u ∣]_{\frac{2}{2}}^{1} : 8 [ln ∣ u ∣]_{\frac{1}{2}}^{1}

= 8 [ln ∣ u ∣]_{\frac{1}{2}}^{1}

Compute the boundaries:

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

= 8 \cdot \frac{1}{2} ln (2)

Simplify

= 4 ln (2)

\frac{d}{d x} (2 x^{3} + x)

\frac{\partial}{\partial x} (y^{2} e^{2 x})

\frac{d}{d x} (e^{x^{2} - 4} + 2 x + 9)

\frac{d}{d x} (tan^{4} (cot (3 x^{4} + 5 x)))

x \frac{d y}{d x} = x^{3} + 24 x^{3} y

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