What is the integral of tan^3(x)+1 ?

The integral of tan^3(x)+1 is -ln|sec(x)|+(sec^2(x))/2+x+C

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integral of tan^3(x)+1

\int tan^{3} (x) + 1 d x

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

Solution steps

\int tan^{3} (x) + 1 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 1 d x

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

\int 1 d x = x

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

Add a constant to the solution

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

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What is the integral of tan^3(x)+1 ?
The integral of tan^3(x)+1 is -ln|sec(x)|+(sec^2(x))/2+x+C