What is the integral from 0 to pi/3 of 1/(1-sin(x)) ?

The integral from 0 to pi/3 of 1/(1-sin(x)) is 1+sqrt(3)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to pi/3 of 1/(1-sin(x))

\int_{0}^{\frac{π}{3}} \frac{1}{1 - sin ( x )} d x

1 + 3

Decimal

2.73205 \dots

Solution steps

\int_{0}^{\frac{π}{3}} \frac{1}{1 - sin ( x )} d x

Apply u-substitution

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

Take the constant out:

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

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

Complete the square

1 + u^{2} - 2 u : (u - 1)^{2}

= 2 \cdot \int_{0}^{\frac{3}{3}} \frac{1}{( u - 1 ) ^{2}} d u

Apply u-substitution

= 2 \cdot \int_{- 1}^{\frac{1}{3} - 1} \frac{1}{v ^{2}} d v

Apply the Power Rule

= 2 [- \frac{1}{v}]_{- 1}^{\frac{1}{3} - 1}

Compute the boundaries:

\frac{3 + 3}{2} - 1

= 2 (\frac{3 + 3}{2} - 1)

Simplify

= 1 + 3

What is the integral from 0 to pi/3 of 1/(1-sin(x)) ?
The integral from 0 to pi/3 of 1/(1-sin(x)) is 1+sqrt(3)