What is the integral of (5tan(x)-cos^2(x))/(cos(x)) ?

The integral of (5tan(x)-cos^2(x))/(cos(x)) is 5sec(x)-sin(x)+C

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integral of (5tan(x)-cos^2(x))/(cos(x))

\int \frac{5 tan ( x ) - cos ^{2} ( x )}{cos ( x )} d x

5 sec (x) - sin (x) + C

Solution steps

\int \frac{5 tan ( x ) - cos ^{2} ( x )}{cos ( x )} d x

Expand

\frac{5 tan ( x ) - cos ^{2} ( x )}{cos ( x )} : \frac{5 tan ( x )}{cos ( x )} - cos (x)

= \int \frac{5 tan ( x )}{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{5 tan ( x )}{cos ( x )} d x - \int cos (x) d x

\int \frac{5 tan ( x )}{cos ( x )} d x = 5 sec (x)

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

= 5 sec (x) - sin (x)

Add a constant to the solution

= 5 sec (x) - sin (x) + C

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

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\int \frac{4}{4 x ^{2} + 1} d x

What is the integral of (5tan(x)-cos^2(x))/(cos(x)) ?
The integral of (5tan(x)-cos^2(x))/(cos(x)) is 5sec(x)-sin(x)+C