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Popular >

integral of (sec(x))/(1-tan(x))

Solution

\int \frac{sec ( x )}{1 - tan ( x )} d x

Solution

2 (\frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} + 1 - \frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} - 1) + C

Solution steps

\int \frac{sec ( x )}{1 - tan ( x )} d x

Express with sin, cos

= \int \frac{1}{cos ( x ) - sin ( x )} d x

Apply u-substitution

: \int \frac{2}{1 - u ^{2} - 2 u} d u

= \int \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 \frac{1}{1 - u ^{2} - 2 u} d u

Complete the square

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

= 2 \cdot \int \frac{1}{- ( u + 1 ) ^{2} + 2} d u

Apply u-substitution

: \int \frac{1}{- v ^{2} + 2} d v

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

Apply Integral Substitution

= 2 \cdot \int \frac{1}{2 ( - w ^{2} + 1 )} d w

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{- w ^{2} + 1} d w

Use the common integral:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= 2 \cdot \frac{1}{2} (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2})

Substitute back

= 2 \cdot \frac{1}{2} \frac{ln \frac{t a n ( \frac{x}{2} ) + 1}{2} + 1}{2} - \frac{ln \frac{t a n ( \frac{x}{2} ) + 1}{2} - 1}{2}

Simplify

2 \cdot \frac{1}{2} \frac{ln \frac{t a n ( \frac{x}{2} ) + 1}{2} + 1}{2} - \frac{ln \frac{t a n ( \frac{x}{2} ) + 1}{2} - 1}{2} : 2 (\frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} + 1 - \frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} - 1)

= 2 (\frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} + 1 - \frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} - 1)

Add a constant to the solution

= 2 (\frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} + 1 - \frac{1}{2} ln \frac{tan ( \frac{x}{2} ) + 1}{2} - 1) + C

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