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

(1.1) (2.4)

\int \frac{6 arcsin ( x )}{x ^{2}} d x

\frac{1}{( 7 + x ) ^{3}}

m = 1 \sum \infty 3 \cdot (\frac{1}{3})

\int (x - 3)^{2} sin (x - 3) d x

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