What is the integral of 5(sec^2(x)-sin(x)) ?

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

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integral of 5(sec^2(x)-sin(x))

\int 5 (sec^{2} (x) - sin (x)) d x

5 (tan (x) + cos (x)) + C

Solution steps

\int 5 (sec^{2} (x) - sin (x)) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int sec^{2} (x) - sin (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

= 5 (\int sec^{2} (x) d x - \int sin (x) d x)

\int sec^{2} (x) d x = tan (x)

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

= 5 (tan (x) - (- cos (x)))

Simplify

= 5 (tan (x) + cos (x))

Add a constant to the solution

= 5 (tan (x) + cos (x)) + C

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