What is the integral of (tan^3(x)+tan(x)) ?

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

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integral of (tan^3(x)+tan(x))

\int (tan^{3} (x) + tan (x)) d x

- ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2} - ln ∣ cos (x) ∣ + C

Solution steps

\int tan^{3} (x) + tan (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 tan^{3} (x) d x + \int tan (x) d x

\int tan^{3} (x) d x = - ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2}

\int tan (x) d x = - ln ∣ cos (x) ∣

= - ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2} - ln ∣ cos (x) ∣

Add a constant to the solution

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

\int \frac{x ^{3}}{x ^{2} + 7} d x

f (0.6) = 3 arcsin (x^{4})

\int y e^{x^{2}} d x

\frac{\partial}{\partial x} (\frac{3 x ^{2}}{y - 5})

\frac{\partial}{\partial z} (\frac{x + y}{z})

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