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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Expand

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

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

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

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

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

Simplify

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

Add a constant to the solution

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

\int_{0}^{\frac{π}{4}} cos^{4} (x) d x

25 x^{4} + 30 x^{3} - 11 x^{2} - 12 x + 4

(5 t + 3 y + 6) d t - d y = 0

y^{^{'}} - 3 y = 9 x

(- 3, - 1), (8, 2)

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