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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int \frac{sin ( x )}{cos ( x )} d x = - ln ∣ cos (x) ∣

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

Add a constant to the solution

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

\int_{0}^{9} x^{2} + 81 d x

\int \frac{( x - 2 ) ^{4}}{4 x - x ^{2}} d x

y = ∣ 10 x ∣, y = x^{2} - 11

\frac{d x}{d t} = 2 \frac{x}{t} - 0.194 t

\frac{\partial}{\partial x} (ln (x^{6} - y^{6}))

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