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

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

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integral from pi/6 to pi/3 of 2cot(x)

\int_{\frac{π}{6}}^{\frac{π}{3}} 2 cot (x) d x

ln (3)

Decimal

1.09861 \dots

Solution steps

\int_{\frac{π}{6}}^{\frac{π}{3}} 2 cot (x) 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}{6}}^{\frac{π}{3}} cot (x) d x

Rewrite using trig identities

= 2 \cdot \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{cos ( x )}{sin ( x )} d x

Apply u-substitution

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

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

Compute the boundaries:

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

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

Simplify

= ln (3)

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\int_{0}^{\frac{t}{2}} t d t

\int_{1}^{3} 95 + 972 t^{2} - 18 t d t

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