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

The integral of x/(1-sin(x)) is xtan(x)+ln|cos(x)|+xsin(x)tan(x)+xcos(x)-ln|tan(x)+sec(x)|+C

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

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

x tan (x) + ln ∣ cos (x) ∣ + x sin (x) tan (x) + x cos (x) - ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Multiply by the conjugate of

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

= \int \frac{x}{\frac{c o s ^{2} ( x )}{1 + s i n ( x )}} d x

Expand

\frac{x}{\frac{c o s ^{2} ( x )}{1 + s i n ( x )}} : \frac{x}{cos ^{2} ( x )} + \frac{x sin ( x )}{cos ^{2} ( x )}

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

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

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

= x tan (x) + ln ∣ cos (x) ∣ + x sin (x) tan (x) + x cos (x) - ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= x tan (x) + ln ∣ cos (x) ∣ + x sin (x) tan (x) + x cos (x) - ln ∣ tan (x) + sec (x) ∣ + C

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What is the integral of x/(1-sin(x)) ?
The integral of x/(1-sin(x)) is xtan(x)+ln|cos(x)|+xsin(x)tan(x)+xcos(x)-ln|tan(x)+sec(x)|+C