What is the integral of 2^x-tan(x)+3cos(x)-(20)/x ?

The integral of 2^x-tan(x)+3cos(x)-(20)/x is (2^x)/(ln(2))+ln|cos(x)|+3sin(x)-20ln|x|+C

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integral of 2^x-tan(x)+3cos(x)-(20)/x

\int 2^{x} - tan (x) + 3 cos (x) - \frac{20}{x} d x

\frac{2 ^{x}}{ln ( 2 )} + ln ∣ cos (x) ∣ + 3 sin (x) - 20 ln ∣ x ∣ + C

Solution steps

\int 2^{x} - tan (x) + 3 cos (x) - \frac{20}{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 2^{x} d x - \int tan (x) d x + \int 3 cos (x) d x - \int \frac{20}{x} d x

\int 2^{x} d x = \frac{2 ^{x}}{ln ( 2 )}

\int tan (x) d x = - ln ∣ cos (x) ∣

\int 3 cos (x) d x = 3 sin (x)

\int \frac{20}{x} d x = 20 ln ∣ x ∣

= \frac{2 ^{x}}{ln ( 2 )} - (- ln ∣ cos (x) ∣) + 3 sin (x) - 20 ln ∣ x ∣

Simplify

= \frac{2 ^{x}}{ln ( 2 )} + ln ∣ cos (x) ∣ + 3 sin (x) - 20 ln ∣ x ∣

Add a constant to the solution

= \frac{2 ^{x}}{ln ( 2 )} + ln ∣ cos (x) ∣ + 3 sin (x) - 20 ln ∣ x ∣ + C

What is the integral of 2^x-tan(x)+3cos(x)-(20)/x ?
The integral of 2^x-tan(x)+3cos(x)-(20)/x is (2^x)/(ln(2))+ln|cos(x)|+3sin(x)-20ln|x|+C