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

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

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

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

- \frac{π}{3} + 3

Decimal

0.68485 \dots

Solution steps

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

Rewrite using trig identities

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

Apply the Sum Rule:

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

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

\int_{0}^{\frac{π}{3}} 1 d x = \frac{π}{3}

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

= - \frac{π}{3} + 3

y = 2 x^{3} - x^{2} + 2, (1, 3)

\int 1 - 4 x^{2} d x

x \to 0 lim (\frac{1}{x} - \frac{1}{x ^{2} + 1})

\int \frac{1}{1 - 25 x ^{2}} d x

f (x) = \frac{x ^{3} + x ^{2} + 6}{x}, at x = 1

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