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∫cot⁡3xcsc⁡5x dx\int \cot^{3}x \csc^{5}x \, dx

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Full solution

Q. ∫cot⁡3xcsc⁡5x dx\int \cot^{3}x \csc^{5}x \, dx
  1. Rewrite Trig Functions: Rewrite the integral in terms of sine and cosine to simplify the expression. \newlinecot⁡(x)=cos⁡(x)sin⁡(x)\cot(x) = \frac{\cos(x)}{\sin(x)} and csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}, so cot⁡3(x)csc⁡5(x)=(cos⁡3(x)sin⁡3(x))(1sin⁡5(x))=cos⁡3(x)sin⁡8(x)\cot^3(x)\csc^5(x) = \left(\frac{\cos^3(x)}{\sin^3(x)}\right)\left(\frac{1}{\sin^5(x)}\right) = \frac{\cos^3(x)}{\sin^8(x)}.\newlineThe integral becomes ∫cos⁡3(x)sin⁡8(x) dx\int \frac{\cos^3(x)}{\sin^8(x)}\,dx.
  2. Use Substitution: Use the substitution u=sin⁡(x)u = \sin(x), which implies du=cos⁡(x)dxdu = \cos(x)dx. This substitution will simplify the integral by removing the trigonometric functions.
  3. Rewrite in terms of uu: Rewrite the integral in terms of uu. The integral becomes ∫(cos⁡3(x)/u8)cos⁡(x) dx\int(\cos^3(x)/u^8)\cos(x)\,dx, which simplifies to ∫(1/u8) du\int(1/u^8)\,du after substitution.
  4. Integrate with u: Integrate with respect to uu. The integral of 1u8\frac{1}{u^8} with respect to uu is −17u7+C-\frac{1}{7u^7} + C, where CC is the constant of integration.
  5. Substitute back to x: Substitute back in terms of x.\newlineSince u=sin⁡(x)u = \sin(x), the integral becomes −17sin⁡7(x)+C-\frac{1}{7\sin^7(x)} + C.
  6. Check for Errors: Check for any mathematical errors. The substitution was done correctly, the integral was computed correctly, and the back-substitution was also correct.

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