What is the integral of cot^4(xcs)c^2x ?

The integral of cot^4(xcs)c^2x is

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integral of cot^4(xcs)c^2x

\int cot^{4} (x cs) c^{2} x d x

- \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x) + C

Solution steps

\int cot^{4} (x cs) c^{2} x d x

Take the constant out:

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

= c^{2} \cdot \int cot^{4} (cs x) x d x

Apply u-substitution

: \int \frac{u cot ^{4} ( u )}{c ^{2} s ^{2}} d u

= c^{2} \cdot \int \frac{u cot ^{4} ( u )}{c ^{2} s ^{2}} d u

Take the constant out:

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

= c^{2} \frac{1}{c ^{2} s ^{2}} \cdot \int u cot^{4} (u) d u

Apply Integration By Parts

: u (- \frac{1}{3} cot^{3} (u) + u + cot (u)) - \int - \frac{1}{3} cot^{3} (u) + u + cot (u) d u

= c^{2} \frac{1}{c ^{2} s ^{2}} (u (- \frac{1}{3} cot^{3} (u) + u + cot (u)) - \int - \frac{1}{3} cot^{3} (u) + u + cot (u) d u)

\int - \frac{1}{3} cot^{3} (u) + u + cot (u) d u = \frac{u ^{2}}{2} + \frac{4}{3} ln ∣ sin (u) ∣ + \frac{1}{6} cot^{2} (u)

= c^{2} \frac{1}{c ^{2} s ^{2}} (u (- \frac{1}{3} cot^{3} (u) + u + cot (u)) - (\frac{u ^{2}}{2} + \frac{4}{3} ln ∣ sin (u) ∣ + \frac{1}{6} cot^{2} (u)))

Substitute back

u = cs x

= c^{2} \frac{1}{c ^{2} s ^{2}} (cs x (- \frac{1}{3} cot^{3} (cs x) + cs x + cot (cs x)) - (\frac{( cs x ) ^{2}}{2} + \frac{4}{3} ln ∣ sin (cs x) ∣ + \frac{1}{6} cot^{2} (cs x)))

Simplify

c^{2} \frac{1}{c ^{2} s ^{2}} (cs x (- \frac{1}{3} cot^{3} (cs x) + cs x + cot (cs x)) - (\frac{( cs x ) ^{2}}{2} + \frac{4}{3} ln ∣ sin (cs x) ∣ + \frac{1}{6} cot^{2} (cs x))) : - \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x)

= - \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x)

Add a constant to the solution

= - \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x) + C

\int cot^{3} (θ) csc^{2} (θ) d θ

\frac{d ^{2} y}{d t ^{2}} + 3 (\frac{d y}{d t}) + 2 y = 1

\int_{1}^{\infty} \frac{x}{6 + x ^{2}} d x

\frac{d}{d x} (0.001 x^{3} - 0.14 x^{2} + 7 x + 160)

[(z - 1)^{3} (z + 1)^{2}] - 1