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)

integral

tan^{3} (x) + tan (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

2 e^{t}

sec^{2} (x) - 1

\frac{2 x}{( 1 + x ^{2} ) ^{2}}

x + 8

lo g_{10} (3 x - 2)

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