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

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

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

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

5 sec (x) - 2 sin (x) + C

Solution steps

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

Expand

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

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

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

\int 2 cos (x) d x = 2 sin (x)

= 5 sec (x) - 2 sin (x)

Add a constant to the solution

= 5 sec (x) - 2 sin (x) + C

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