What is the integral from pi/(12) to pi/6 of csc(3x) ?

The integral from pi/(12) to pi/6 of csc(3x) is -1/3 ln(sqrt(2)-1)

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integral from pi/(12) to pi/6 of csc(3x)

\int_{\frac{π}{12}}^{\frac{π}{6}} csc (3 x) d x

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

Decimal

0.29379 \dots

Solution steps

\int_{\frac{π}{12}}^{\frac{π}{6}} csc (3 x) d x

Apply u-substitution

: \int_{\frac{π}{4}}^{\frac{π}{2}} csc (u) \frac{1}{3} d u

= \int_{\frac{π}{4}}^{\frac{π}{2}} csc (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{\frac{π}{4}}^{\frac{π}{2}} csc (u) d u

Apply u-substitution

: \int_{2 - 1}^{1} \frac{1}{v} d v

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} [ln ∣ v ∣]_{2 - 1}^{1}

Compute the boundaries:

- ln (2 - 1)

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

Simplify

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

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What is the integral from pi/(12) to pi/6 of csc(3x) ?
The integral from pi/(12) to pi/6 of csc(3x) is -1/3 ln(sqrt(2)-1)