What is the integral of (-1)/(1+sin(x)) ?

The integral of (-1)/(1+sin(x)) is 2/(tan(x/2)+1)+C

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integral of (-1)/(1+sin(x))

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

What is the integral of (-1)/(1+sin(x)) ?
The integral of (-1)/(1+sin(x)) is 2/(tan(x/2)+1)+C