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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

3 - \frac{π}{3}

Decimal Notation

0.68485 \dots

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{1 - cos ^{2} ( x )}{cos ^{2} ( x )} : \frac{1}{cos ^{2} ( x )} - 1

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

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

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

= 3 - \frac{π}{3}

\int 9 ln (x^{2} - 1) d x

y^{^{'}} + \frac{y}{x} = \frac{y ^{2}}{x}

\frac{d}{d x} (x^{4} - 5 x^{2} + 4)

y = \frac{4}{( 4 - x ^{4} ) ^{\frac{3}{2}}}

\int_{0}^{\frac{π}{2}} 9 (cos (x))^{2} d x

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