What is the integral from 0 to pi/3 of 6sec^3(x) ?

The integral from 0 to pi/3 of 6sec^3(x) is 6(sqrt(3)+1/2 ln(2+sqrt(3)))

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integral from 0 to pi/3 of 6sec^3(x)

\int_{0}^{\frac{π}{3}} 6 sec^{3} (x) d x

6 (3 + \frac{1}{2} ln (2 + 3))

Decimal

14.34317 \dots

Solution steps

\int_{0}^{\frac{π}{3}} 6 sec^{3} (x) d x

Take the constant out:

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

= 6 \cdot \int_{0}^{\frac{π}{3}} sec^{3} (x) d x

Apply Integral Reduction

= 6 ([\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)

= 6 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3))

Simplify

6 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)) : 6 ([\frac{1}{2} sec (x) tan (x)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3))

= 6 ([\frac{1}{2} sec (x) tan (x)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3))

Compute the boundaries:

3

= 6 (3 + \frac{1}{2} ln (2 + 3))

x \to - 0.001 lim (\frac{e ^{x} - 1}{x})

\int_{- 2}^{3} x^{3} d x

\int \frac{1}{( x ^{2} - 1 ) ^{3}} d x

x \to 3 lim (4 x^{2})

\frac{\partial}{\partial x} (a cos (x) sin (y))

What is the integral from 0 to pi/3 of 6sec^3(x) ?
The integral from 0 to pi/3 of 6sec^3(x) is 6(sqrt(3)+1/2 ln(2+sqrt(3)))