What is the integral of 4tan^2(x)+tan^4(x) ?

The integral of 4tan^2(x)+tan^4(x) is -3x+1/3 tan^3(x)+3tan(x)+C

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integral of 4tan^2(x)+tan^4(x)

\int 4 tan^{2} (x) + tan^{4} (x) d x

- 3 x + \frac{1}{3} tan^{3} (x) + 3 tan (x) + C

Solution steps

\int 4 tan^{2} (x) + tan^{4} (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 4 tan^{2} (x) d x + \int tan^{4} (x) d x

\int 4 tan^{2} (x) d x = 4 (- x + tan (x))

\int tan^{4} (x) d x = \frac{1}{3} tan^{3} (x) + x - tan (x)

= 4 (- x + tan (x)) + \frac{1}{3} tan^{3} (x) + x - tan (x)

Simplify

4 (- x + tan (x)) + \frac{1}{3} tan^{3} (x) + x - tan (x) : - 3 x + \frac{1}{3} tan^{3} (x) + 3 tan (x)

= - 3 x + \frac{1}{3} tan^{3} (x) + 3 tan (x)

Add a constant to the solution

= - 3 x + \frac{1}{3} tan^{3} (x) + 3 tan (x) + C

What is the integral of 4tan^2(x)+tan^4(x) ?
The integral of 4tan^2(x)+tan^4(x) is -3x+1/3 tan^3(x)+3tan(x)+C