What is the integral of sec^6(t) ?

The integral of sec^6(t) is tan(t)+(2tan^3(t))/3+(tan^5(t))/5+C

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integral of sec^6(t)

\int sec^{6} (t) d t

tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5} + C

Solution steps

\int sec^{6} (t) d t

Rewrite using trig identities

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

Apply u-substitution

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

Expand

(1 + u^{2})^{2} : 1 + 2 u^{2} + u^{4}

= \int 1 + 2 u^{2} + u^{4} 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 2 u^{2} d u + \int u^{4} d u

\int 1 d u = u

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

\int u^{4} d u = \frac{u ^{5}}{5}

= u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}

Substitute back

u = tan (t)

= tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5}

Add a constant to the solution

= tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5} + C

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What is the integral of sec^6(t) ?
The integral of sec^6(t) is tan(t)+(2tan^3(t))/3+(tan^5(t))/5+C