What is the integral of (1-sin(x))/(1-sin^2(x)) ?

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

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integral of (1-sin(x))/(1-sin^2(x))

\int \frac{1 - sin ( x )}{1 - sin ^{2} ( x )} d x

tan (x) - sec (x) + C

Solution steps

\int \frac{1 - sin ( x )}{1 - sin ^{2} ( x )} d x

Rewrite using trig identities

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

Expand

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

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

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

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

= tan (x) - sec (x)

Add a constant to the solution

= tan (x) - sec (x) + C

y = x^{2} + 3, (- 1, 4)

\int_{- 4}^{4} ∣ x ∣ d x

x \to - 3 lim (\frac{4 x - 5}{3 x + 9})

x \to 1 lim ([3 x + 1])

4 x

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