What is the integral of (sec^4(x))/(tan(x)) ?

The integral of (sec^4(x))/(tan(x)) is ln|tan(x)|+(tan^2(x))/2+C

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integral of (sec^4(x))/(tan(x))

\int \frac{sec ^{4} ( x )}{tan ( x )} d x

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

Solution steps

\int \frac{sec ^{4} ( x )}{tan ( x )} d x

Apply Trigonometric Substitution

: \int \frac{1 + u ^{2}}{u} d u

= \int \frac{1 + u ^{2}}{u} d u

Expand

\frac{1 + u ^{2}}{u} : \frac{1}{u} + u

= \int \frac{1}{u} + u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u + \int u d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

= ln ∣ u ∣ + \frac{u ^{2}}{2}

Substitute back

u = tan (x)

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

Add a constant to the solution

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

y = x^{3} - 6 x, (0, 3)

\int \frac{24}{25 + r ^{2}} d r

\int 1 - sin (x) d x

\int_{1}^{3} x^{2} - 4 x + 3 d x

f (x) = e^{x} + sin (x)

What is the integral of (sec^4(x))/(tan(x)) ?
The integral of (sec^4(x))/(tan(x)) is ln|tan(x)|+(tan^2(x))/2+C