Question

Tried doing it myself multiple times , multi choice question, not getting any availabile answer.

Answer 1

Answer: The given integral is impossible to solve.

Step-by-step explanation: I tried solving it right now with integration by parts and trigonometric substitution, and I was unable to solve it.

So I looked at two math solvers and tried inputting the question, and it either says "There is nothing more you can do with this problem," or "We are unable to solve this problem."

Answer 2

First, let's substitute $x=\frac w4$ and $dx=\frac{dw}4$ to get rid of the fraction.

$\displaystyle \int \csc^6\left(\dfrac w4\right) \cot^4\left(\dfrac w4\right) \, dw = 4 \int \csc^6(x) \cot^4(x) \, dx$

Recall that

$\cot^2(x) + 1 = \csc^2(x)$

$\dfrac{d}{dx} \cot(x) = -\csc^2(x)$

We can the rewrite the integrand as

$\displaystyle 4 \int \csc^6(x) \cot^4(x) \, dx = 4 \int \left(\cot^2(x) + 1\right)^2 \cot^4(x) \csc^2(x) \, dx$

then substitute $y=\cot(x)$ and $dy=-\csc^2(x)\,dx$ to get

$\displaystyle -4 \int (y^2 + 1)^2 y^4 \, dy$

Expand the integrand.

$\displaystyle -4 \int (y^4 + 2y^2 + 1) y^4 \, dy = -4 \int (y^8 + 2y^6 + y^4) \, dy$

Now integrate with the power rule.

$\displaystyle -4 \left(\frac{y^9}9 + \frac{2y^7}7 + \frac{y^5}5\right) + C$

$\displaystyle -\frac{4y^9}9 - \frac{8y^7}7 - \frac{4y^5}5 + C$

Put everything back in terms of $x$, then $w$.

$\displaystyle -\frac{4\cot^9(x)}9 - \frac{8\cot^7(x)}7 - \frac{4\cot^5(x)}5 + C$

$\displaystyle \boxed{-\frac49 \cot^9\left(\dfrac w4\right) - \frac87 \cot^7\left(\dfrac w4\right) - \frac45 \cot^5\left(\dfrac w4\right) + C}$