int3cosec^(5)x cot xdx

Recognize Trigonometric Functions: Given the integral to solve is ∫3cosec^5(x) cot(x) dx. We can start by recognizing that cosec(x) is 1/sin(x) and cot(x) is cos(x)/sin(x). So, we rewrite the integral in terms of sine and cosine: ∫3cosec^5(x) cot(x) dx = ∫3(1/sin^5(x)) * (cos(x)/sin(x)) dx = ∫3cos(x)/sin^6(x) dx Now, let's use a substitution method where we let u = sin(x), which means du = cos(x) dx.Perform Substitution: Perform the substitution: u = sin(x) => du = cos(x) dx The integral becomes: ∫3cos(x)/sin^6(x) dx = ∫3/u^6 duIntegrate with Respect to u: Now we integrate 3/u^6 with respect to u: ∫3/u^6 du = 3∫u^(-6) du The antiderivative of u^(-6) is u^(-5)/(-5). So, we have 3 * u^(-5)/(-5) + C, where C is the constant of integration.Substitute Back for u: Substitute back for u to get the answer in terms of x: 3 * u^(-5)/(-5) + C = 3 * sin^(-5)(x)/(-5) + C Simplify the expression: = -3/5 * sin^(-5)(x) + C = -3/5 * cosec^5(x) + C This is our final answer.

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\int 3 \operatorname{cosec}^{5} x \cot x d x

Recognize Trigonometric Functions: Given the integral to solve is $\int 3 \csc^5(x) \cot(x) \, dx$ . We can start by recognizing that $\csc(x)$ is $\frac{1}{\sin(x)}$ and $\cot(x)$ is $\frac{\cos(x)}{\sin(x)}$ . So, we rewrite the integral in terms of sine and cosine: $\int 3 \csc^5(x) \cot(x) \, dx = \int 3\left(\frac{1}{\sin^5(x)}\right) \cdot \left(\frac{\cos(x)}{\sin(x)}\right) \, dx = \int \frac{3\cos(x)}{\sin^6(x)} \, dx$ Now, let's use a substitution method where we let $u = \sin(x)$ , which means $du = \cos(x) \, dx$ .
Perform Substitution: Perform the substitution: $\newline$ $u = \sin(x) \Rightarrow du = \cos(x) dx$ $\newline$ The integral becomes: $\newline$ $\int 3\cos(x)/\sin^6(x) dx = \int 3/u^6 du$
Integrate with Respect to u: Now we integrate $\frac{3}{u^6}$ with respect to $u$ : $\newline$ $\int \frac{3}{u^6} du = 3\int u^{-6} du$ $\newline$ The antiderivative of $u^{-6}$ is $\frac{u^{-5}}{-5}$ . $\newline$ So, we have $\frac{3 \cdot u^{-5}}{-5} + C$ , where $C$ is the constant of integration.
Substitute Back for u: Substitute back for u to get the answer in terms of x: $\newline$ $3 \cdot u^{-5}/(-5) + C = 3 \cdot \sin^{-5}(x)/(-5) + C$ $\newline$ Simplify the expression: $\newline$ $= -\frac{3}{5} \cdot \sin^{-5}(x) + C$ $\newline$ $= -\frac{3}{5} \cdot \csc^5(x) + C$ $\newline$ This is our final answer.