What is the integral of tan^3(t)sec(t) ?

The integral of tan^3(t)sec(t) is -sec(t)+(sec^3(t))/3+C

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integral of tan^3(t)sec(t)

\int tan^{3} (t) sec (t) d t

- sec (t) + \frac{sec ^{3} ( t )}{3} + C

Solution steps

\int tan^{3} (t) sec (t) d t

Rewrite using trig identities

= \int (- 1 + sec^{2} (t)) tan (t) sec (t) d t

Apply u-substitution

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

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= - u + \frac{u ^{3}}{3}

Substitute back

u = sec (t)

= - sec (t) + \frac{sec ^{3} ( t )}{3}

Add a constant to the solution

= - sec (t) + \frac{sec ^{3} ( t )}{3} + C

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What is the integral of tan^3(t)sec(t) ?
The integral of tan^3(t)sec(t) is -sec(t)+(sec^3(t))/3+C