What is the integral of 5cot^3(x)csc^3(x) ?

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

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integral of 5cot^3(x)csc^3(x)

\int 5 cot^{3} (x) csc^{3} (x) d x

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

Solution steps

\int 5 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

= 5 \cdot \int cot^{3} (x) csc^{3} (x) d x

Apply u-substitution

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

= 5 \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

= 5 (- \int u^{2} (u^{2} - 1) d u)

Expand

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

= 5 (- \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

= 5 (- (\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}

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

Substitute back

u = csc (x)

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

Simplify

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

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

Add a constant to the solution

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

ln ∣ sec (x) ∣

\int sec (5 x) d x

f (x) = 7 e^{x}, at x = e

y^{^{''}} + 10 y^{^{'}} + 29 y = 0

x y \frac{d y}{d x} = x^{2} + 3 y^{2}

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