What is the integral of (tan^2(x)+cot^3(x)) ?

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

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integral of (tan^2(x)+cot^3(x))

\int (tan^{2} (x) + cot^{3} (x)) d x

- x + tan (x) - \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣ + C

Solution steps

\int tan^{2} (x) + cot^{3} (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^{2} (x) d x + \int cot^{3} (x) d x

\int tan^{2} (x) d x = - x + tan (x)

\int cot^{3} (x) d x = - \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣

= - x + tan (x) - \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣

Add a constant to the solution

= - x + tan (x) - \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣ + C

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