What is the integral from 0 to pi/4 of 1/(1-sin(θ)) ?

The integral from 0 to pi/4 of 1/(1-sin(θ)) is square root of 2

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integral from 0 to pi/4 of 1/(1-sin(θ))

\int_{0}^{\frac{π}{4}} \frac{1}{1 - sin ( θ )} d θ

2

Decimal Notation

1.41421 \dots

Solution steps

\int_{0}^{\frac{π}{4}} \frac{1}{1 - sin ( θ )} d θ

Apply u-substitution

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

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

Complete the square

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

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

Apply u-substitution

: \int_{- 1}^{2 - 2} \frac{1}{v ^{2}} d v

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

Apply the Power Rule

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

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

Compute the boundaries:

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

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

Simplify

= 2

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What is the integral from 0 to pi/4 of 1/(1-sin(θ)) ?
The integral from 0 to pi/4 of 1/(1-sin(θ)) is square root of 2