What is the integral from 0 to pi/6 of 20sec^4(x) ?

The integral from 0 to pi/6 of 20sec^4(x) is (200)/(9sqrt(3))

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integral from 0 to pi/6 of 20sec^4(x)

\int_{0}^{\frac{π}{6}} 20 sec^{4} (x) d x

\frac{200}{9 3}

Decimal

12.83000 \dots

Solution steps

\int_{0}^{\frac{π}{6}} 20 sec^{4} (x) d x

Take the constant out:

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

= 20 \cdot \int_{0}^{\frac{π}{6}} sec^{4} (x) d x

Rewrite using trig identities

= 20 \cdot \int_{0}^{\frac{π}{6}} (1 + tan^{2} (x)) sec^{2} (x) d x

Apply u-substitution

= 20 \cdot \int_{0}^{\frac{3}{3}} 1 + u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 20 (\int_{0}^{\frac{3}{3}} 1 d u + \int_{0}^{\frac{3}{3}} u^{2} d u)

\int_{0}^{\frac{3}{3}} 1 d u = \frac{1}{3}

\int_{0}^{\frac{3}{3}} u^{2} d u = \frac{1}{9 3}

= 20 (\frac{1}{3} + \frac{1}{9 3})

Simplify

20 (\frac{1}{3} + \frac{1}{9 3}) : \frac{200}{9 3}

= \frac{200}{9 3}

What is the integral from 0 to pi/6 of 20sec^4(x) ?
The integral from 0 to pi/6 of 20sec^4(x) is (200)/(9sqrt(3))