What is the integral of 3cot^3(x)csc^3(x) ?

The integral of 3cot^3(x)csc^3(x) is -3/5 csc^5(x)+csc^3(x)+C

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integral of 3cot^3(x)csc^3(x)

\int 3 cot^{3} (x) csc^{3} (x) d x

- \frac{3}{5} csc^{5} (x) + csc^{3} (x) + C

Solution steps

\int 3 cot^{3} (x) csc^{3} (x) d x

Take the constant out:

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

= 3 \cdot \int cot^{3} (x) csc^{3} (x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Expand

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

= 3 (- \int u^{4} - 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 u^{4} d u - \int u^{2} d u))

\int u^{4} d u = \frac{u ^{5}}{5}

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

= 3 (- (\frac{u ^{5}}{5} - \frac{u ^{3}}{3}))

Substitute back

u = csc (x)

= 3 (- (\frac{csc ^{5} ( x )}{5} - \frac{csc ^{3} ( x )}{3}))

Simplify

3 (- (\frac{csc ^{5} ( x )}{5} - \frac{csc ^{3} ( x )}{3})) : - \frac{3}{5} csc^{5} (x) + csc^{3} (x)

= - \frac{3}{5} csc^{5} (x) + csc^{3} (x)

Add a constant to the solution

= - \frac{3}{5} csc^{5} (x) + csc^{3} (x) + C

\int \frac{5 x - 2}{( x + 2 ) ( x - 2 )} d x

\frac{d y}{d x} = x^{2} (1 + y), y (0) = 3

x \to 0 lim (csc (x) ln (x + 1))

5 + \frac{1}{2} sin (5 π t)

y^{^{''}} - 8 y^{^{'}} + 20 y = 100 x^{2} - 91 x e^{x}

What is the integral of 3cot^3(x)csc^3(x) ?
The integral of 3cot^3(x)csc^3(x) is -3/5 csc^5(x)+csc^3(x)+C