What is the integral of 5sec^6(t) ?

The integral of 5sec^6(t) is 5tan(t)+10/3 tan^3(t)+tan^5(t)+C

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integral of 5sec^6(t)

\int 5 sec^{6} (t) d t

5 tan (t) + \frac{10}{3} tan^{3} (t) + tan^{5} (t) + C

Solution steps

\int 5 sec^{6} (t) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int sec^{6} (t) d t

Rewrite using trig identities

= 5 \cdot \int sec^{2} (t) (1 + tan^{2} (t))^{2} d t

Apply u-substitution

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

= 5 \cdot \int (1 + u^{2})^{2} d u

Expand

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

= 5 \cdot \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

= 5 (\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}

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

Substitute back

u = tan (t)

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

Simplify

5 (tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5}) : 5 tan (t) + \frac{10}{3} tan^{3} (t) + tan^{5} (t)

= 5 tan (t) + \frac{10}{3} tan^{3} (t) + tan^{5} (t)

Add a constant to the solution

= 5 tan (t) + \frac{10}{3} tan^{3} (t) + tan^{5} (t) + C

y = - x^{2} + 5 x + 2

x y^{^{''}} + 3 y^{^{'}} = 1984 x^{1982}

\int \frac{2 e ^{x}}{e ^{2 x} + 6 e ^{x} + 9} d x

f (x) = 4 x^{2} cos (x) - 9 x

\frac{d}{d x} (700 x^{2} - 0.2 x)

What is the integral of 5sec^6(t) ?
The integral of 5sec^6(t) is 5tan(t)+10/3 tan^3(t)+tan^5(t)+C