What is the integral of sec^3(x)csc(x) ?

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

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integral of sec^3(x)csc(x)

\int sec^{3} (x) csc (x) d x

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

Solution steps

\int sec^{3} (x) csc (x) d x

Rewrite using trig identities

= \int \frac{1}{cos ^{3} ( x ) sin ( x )} d x

Apply Trigonometric Substitution

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

Simplify

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

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

Expand

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} + ln ∣ u ∣

Substitute back

u = tan (x)

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

Add a constant to the solution

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

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