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

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

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integral of-(sin(x))/(cos^3(x))

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

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

Solution steps

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

Take the constant out:

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

= - \int \frac{sin ( x )}{cos ^{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

x \to 0 lim ((1 + 5 x)^{\frac{4}{x}})

\frac{d y}{d x} - \frac{y}{x} = 3 x e^{x}, y (1) = 3 e - 2

y = - \frac{5 t}{( t + 8 ) ^{2}}

\int (x)^{3} arccos (x) d x

\int \frac{1}{1 + sin ( 2 x )} d x

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