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

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

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

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

23 + ln (2 + 3)

Decimal

4.78105 \dots

Solution steps

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

Apply Integral Reduction

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

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

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

Simplify

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

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

Compute the boundaries:

3

= 23 + ln (2 + 3)

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