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

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

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integral of (tan^2(x)+sec^2(x))/(tan(x))

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

- ln ∣ cos (x) ∣ + ln ∣ tan (x) ∣ + C

Solution steps

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

Expand

\frac{tan ^{2} ( x ) + sec ^{2} ( x )}{tan ( x )} : tan (x) + \frac{sec ^{2} ( x )}{tan ( x )}

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

\int tan (x) d x = - ln ∣ cos (x) ∣

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

= - ln ∣ cos (x) ∣ + ln ∣ tan (x) ∣

Add a constant to the solution

= - ln ∣ cos (x) ∣ + ln ∣ tan (x) ∣ + C

\int \frac{2 x}{x ^{4} + 1} d x

f (x) = 12 x^{3} - \frac{1}{x}

x^{2} + 2 x + 2

(126 - 0.5 x) x - 50 x - 31.75

1 - \frac{d v}{d x} = cos (v)

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