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

The integral of 1/(sqrt(1+sin(x))) is

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{2}{( u + 1 ) 1 + u ^{2}} d u

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

Apply Trigonometric Substitution

: \int \frac{sec ( v )}{tan ( v ) + 1} d v

= 2 \cdot \int \frac{sec ( v )}{tan ( v ) + 1} d v

Apply u-substitution

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

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

Take the constant out:

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

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

Complete the square

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

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

Apply u-substitution

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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