What is the integral from-pi/3 to 0 of 2sec^3(x) ?

The integral from-pi/3 to 0 of 2sec^3(x) is 2sqrt(3)-ln(2-sqrt(3))

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integral from-pi/3 to 0 of 2sec^3(x)

\int_{- \frac{π}{3}}^{0} 2 sec^{3} (x) d x

23 - ln (2 - 3)

Solution steps

\int_{- \frac{π}{3}}^{0} 2 sec^{3} (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}{3}}^{0} sec^{3} (x) d x

Apply Integral Reduction

= 2 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{- \frac{π}{3}}^{0} + \frac{1}{2} \cdot \int_{- \frac{π}{3}}^{0} sec (x) d x)

\int_{- \frac{π}{3}}^{0} sec (x) d x = - ln (2 - 3)

= 2 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{- \frac{π}{3}}^{0} + \frac{1}{2} (- ln (2 - 3)))

Simplify

2 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{- \frac{π}{3}}^{0} + \frac{1}{2} (- ln (2 - 3))) : 2 [\frac{1}{2} sec (x) tan (x)]_{- \frac{π}{3}}^{0} - ln (2 - 3)

= 2 [\frac{1}{2} sec (x) tan (x)]_{- \frac{π}{3}}^{0} - ln (2 - 3)

Compute the boundaries:

3

= 23 - ln (2 - 3)

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What is the integral from-pi/3 to 0 of 2sec^3(x) ?
The integral from-pi/3 to 0 of 2sec^3(x) is 2sqrt(3)-ln(2-sqrt(3))