What is the integral from 0 to pi/3 of sec^5(x) ?

The integral from 0 to pi/3 of sec^5(x) is 2sqrt(3)+3/4 (sqrt(3)+1/2 ln(2+sqrt(3)))

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integral from 0 to pi/3 of sec^5(x)

\int_{0}^{\frac{π}{3}} sec^{5} (x) d x

23 + \frac{3}{4} (3 + \frac{1}{2} ln (2 + 3))

Decimal Notation

5.25699 \dots

Solution steps

\int_{0}^{\frac{π}{3}} sec^{5} (x) d x

Apply Integral Reduction

= [\frac{sec ^{4} ( x ) sin ( x )}{4}]_{0}^{\frac{π}{3}} + \frac{3}{4} \cdot \int_{0}^{\frac{π}{3}} sec^{3} (x) d x

\int_{0}^{\frac{π}{3}} sec^{3} (x) d x = 3 + \frac{1}{2} ln (2 + 3)

= [\frac{sec ^{4} ( x ) sin ( x )}{4}]_{0}^{\frac{π}{3}} + \frac{3}{4} (3 + \frac{1}{2} ln (2 + 3))

Compute the boundaries:

23

= 23 + \frac{3}{4} (3 + \frac{1}{2} ln (2 + 3))

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What is the integral from 0 to pi/3 of sec^5(x) ?
The integral from 0 to pi/3 of sec^5(x) is 2sqrt(3)+3/4 (sqrt(3)+1/2 ln(2+sqrt(3)))