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

The integral of ((sin(x)+cos(x))^2)/(sin(x)) is 2sin(x)+ln|tan(x/2)|+C

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

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

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

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

\int sin (x) d x = - cos (x)

\int 2 cos (x) d x = 2 sin (x)

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

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

Simplify

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

Add a constant to the solution

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

\frac{1}{z ^{2} + 1}

x^{- 5} (1 + x^{- 1}), at x = 1

y = x^{2}, y^{2} = x

f (x) = 5 x - x^{2}

y^{^{''}} - 10 y + 2 y = 0

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