$$\int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dx}$$
I know this is a simple problem, but I don't have the answer for it and I just want to make sure that I'm correct!
$$\int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dx}$$
I know this is a simple problem, but I don't have the answer for it and I just want to make sure that I'm correct!
$$ \int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dy} = y(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3) + \text{const}. $$
So I'm assuming your integral is actually:
$$ \int (3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3) \color{red}{dx}$$
By trig identities $$\csc(x) \cot(x) = \frac{\cos(x)}{\sin^2(x)}.$$ Now if we use $u = \sin(x)$ then $du = \cos(x)dx,$ and $\sin^2(x) =u^2.$ So $$ \int \frac{\cos(x)}{\sin^2(x)} dx = \int \frac{1}{u^2} du .$$
Can you take it from here?
Edit: the complete integral is then
$$ \int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)\ \text{d}x} = -\csc(x) - \frac{5}{8}x^8 + 4 \log(|x|) + 3x + \text{const}. $$
$$\forall -1\neq n\in\Bbb R\,\,,\,\int x^n\,dx=\frac{x^{n+1}}{n+1}+K\,\,,\,\int x^{-1}\,dx=\log|x|+K$$
$$\int\csc x\cot x\,dx=\int\frac{\cos x\,dx}{\sin^2 x}=\int\frac{d(\sin x)}{(\sin x)^2}=-\frac{1}{\sin x}+K\,$$