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

The integral of-sin(x)sec^3(x) is -(tan^2(x))/2+C

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

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

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

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= - \int tan (x) sec^{2} (x) d x

Apply u-substitution

: \int u d u

= - \int u d u

Apply the Power Rule

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

= - \frac{u ^{2}}{2}

Substitute back

u = tan (x)

= - \frac{tan ^{2} ( x )}{2}

Add a constant to the solution

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

\int \frac{x ^{3} + 3 x ^{2} - 9 x - 2}{x - 2} d x

\int \frac{1 + x}{x} \frac{1}{x} d x

\int ∣ 2 x - 4 ∣ d x

\int \frac{sec ( x )}{2 tan ( x )} d x

What is the integral of-sin(x)sec^3(x) ?
The integral of-sin(x)sec^3(x) is -(tan^2(x))/2+C