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

The integral of-3sec(x)*(sec(x))^3 is -3tan(x)-tan^3(x)+C

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integral of-3sec(x)*(sec(x))^3

integral

- 3 sec (x) \cdot (sec (x))^{3}

- 3 tan (x) - tan^{3} (x) + C

Solution steps

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

Take the constant out:

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

= - 3 \cdot \int sec (x) (sec (x))^{3} d x

Simplify

sec (x) (sec (x))^{3} : sec^{4} (x)

= - 3 \cdot \int sec^{4} (x) d x

Rewrite using trig identities

= - 3 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) d x

Apply u-substitution

= - 3 \cdot \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

= - 3 (\int 1 d u + \int u^{2} d u)

\int 1 d u = u

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

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

Substitute back

u = tan (x)

= - 3 (tan (x) + \frac{tan ^{3} ( x )}{3})

Simplify

- 3 (tan (x) + \frac{tan ^{3} ( x )}{3}) : - 3 tan (x) - tan^{3} (x)

= - 3 tan (x) - tan^{3} (x)

Add a constant to the solution

= - 3 tan (x) - tan^{3} (x) + C

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