What is the integral of (sin^2(x))/(sin(x)+cos(x)) ?

The integral of (sin^2(x))/(sin(x)+cos(x)) is

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integral of (sin^2(x))/(sin(x)+cos(x))

\int \frac{sin ^{2} ( x )}{sin ( x ) + cos ( x )} d x

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

Solution steps

\int \frac{sin ^{2} ( x )}{sin ( x ) + cos ( x )} d x

Rewrite using trig identities

= \int \frac{1 - cos ( 2 x )}{2 ( sin ( x ) + cos ( x ) )} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1 - cos ( 2 x )}{sin ( x ) + cos ( x )} d x

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{1}{sin ( x ) + cos ( x )} d x - \int \frac{cos ( 2 x )}{sin ( x ) + cos ( x )} d x)

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

\int \frac{cos ( 2 x )}{sin ( x ) + cos ( x )} d x = sin (x) + cos (x)

= \frac{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) - (sin (x) + cos (x)))

Simplify

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

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

Add a constant to the solution

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

t \to 0 lim (sin (t))

\frac{d}{d x} (2 x^{\frac{5}{3}} - 5 x^{\frac{4}{5}} + 7 x^{\frac{3}{2}} - 8 x + 17)

\int arcsin^{2} (x) d x

f (x) = \frac{x ^{2}}{x - 11}

x \to \infty lim (\frac{e ^{5 x}}{9 x ^{4} + 31 x + 1})

What is the integral of (sin^2(x))/(sin(x)+cos(x)) ?
The integral of (sin^2(x))/(sin(x)+cos(x)) is