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

The integral of (cos(x))/((sin(x))^3) is -1/2 csc^2(x)+C

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

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

- \frac{1}{2} csc^{2} (x) + C

Solution steps

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

Simplify

sin^{3} (x) : sin^{2} (x) sin (x)

= \int \frac{cos ( x )}{sin ^{2} ( x ) sin ( x )} d x

Rewrite using trig identities

= \int cot (x) csc^{2} (x) d x

Apply u-substitution

: \int - \frac{1}{2} d u

= \int - \frac{1}{2} d u

Integral of a constant:

\int a d x = a x

= (- \frac{1}{2}) u

Substitute back

u = csc^{2} (x)

= (- \frac{1}{2}) csc^{2} (x)

Simplify

= - \frac{1}{2} csc^{2} (x)

Add a constant to the solution

= - \frac{1}{2} csc^{2} (x) + C

x \to - 7 + lim (\frac{( - 6 x )}{x + 7})

\frac{d ^{2}}{d x ^{2}} (ln (x^{2} + 1))

y^{^{''}} + 6 y^{^{'}} + 9 y = 6 cos (t) + 8 sin (t)

x \to 0 + lim ((1 - 8 x)^{\frac{1}{x}})

f (x) = x^{2} ln (7 x)

What is the integral of (cos(x))/((sin(x))^3) ?
The integral of (cos(x))/((sin(x))^3) is -1/2 csc^2(x)+C