What is the integral of (4+5sin(x))/(cos(x)) ?

The integral of (4+5sin(x))/(cos(x)) is 4ln|tan(x)+sec(x)|-5ln|cos(x)|+C

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integral of (4+5sin(x))/(cos(x))

integral

\frac{4 + 5 sin ( x )}{cos ( x )}

4 ln ∣ tan (x) + sec (x) ∣ - 5 ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{4 + 5 sin ( x )}{cos ( x )} d x

Expand

\frac{4 + 5 sin ( x )}{cos ( x )} : \frac{4}{cos ( x )} + \frac{5 sin ( x )}{cos ( x )}

= \int \frac{4}{cos ( x )} + \frac{5 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{4}{cos ( x )} d x + \int \frac{5 sin ( x )}{cos ( x )} d x

\int \frac{4}{cos ( x )} d x = 4 ln ∣ tan (x) + sec (x) ∣

\int \frac{5 sin ( x )}{cos ( x )} d x = - 5 ln ∣ cos (x) ∣

= 4 ln ∣ tan (x) + sec (x) ∣ - 5 ln ∣ cos (x) ∣

Add a constant to the solution

= 4 ln ∣ tan (x) + sec (x) ∣ - 5 ln ∣ cos (x) ∣ + C

\frac{1}{( 3 x )}

\frac{1}{( 1 - x ) ^{n}}

f (x)^{2}

\frac{x}{( 1 + x ) ^{2}}

y sin (y^{2})

What is the integral of (4+5sin(x))/(cos(x)) ?
The integral of (4+5sin(x))/(cos(x)) is 4ln|tan(x)+sec(x)|-5ln|cos(x)|+C