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

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

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

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

ln ∣ 1 + sin (x) ∣ + C

Solution steps

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

Multiply by the conjugate of

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

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

Simplify

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

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 1 + sin (x)

= ln ∣ 1 + sin (x) ∣

Add a constant to the solution

= ln ∣ 1 + sin (x) ∣ + C

\frac{d}{d x} (x + e^{- 2 x})

x^{2} y d x + (x^{3} + 3 x y^{2}) d y = 0

t y^{^{'}} + (1 + t) y = t

y^{^{''}} + 7 y^{^{'}} + 12 y = 2 e^{t}

t^{2} y^{^{''}} - 2 t y^{^{'}} + 2 y = 4 t^{2}

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