Evaluate the indefinite integral. ∫(cot⁡x)ln⁡(sin⁡x)dx

Question

Evaluate the indefinite integral.  ∫(cot⁡x)ln⁡(sin⁡x)dx

Lounctirough · Accepted Answer

Step 1&amp;nbsp;We must assess the integral∫(cot⁡x)ln⁡(sin⁡x)&amp;nbsp;dx&amp;nbsp;...(1)&amp;nbsp;let&amp;nbsp;cot⁡x=z⇒log⁡(sin⁡x)&amp;nbsp;dx&amp;nbsp;=&amp;nbsp;dz&amp;nbsp;&amp;nbsp;Step 2&amp;nbsp;substituting it in the equation (1) we get&amp;nbsp;∫z&amp;nbsp;dz&amp;nbsp;&amp;nbsp;integrating it we get&amp;nbsp;[z22]&amp;nbsp;adding z&#039;s value, we obtain[cot2x2]&amp;nbsp;therefore the answer of&amp;nbsp;∫(cot⁡x)ln⁡(sin⁡x)&amp;nbsp;dx&amp;nbsp;&amp;nbsp;is&amp;nbsp;cot2x2

Marlene Broomfield · Accepted Answer

Step 1: Remove parentheses.  ∫cot⁡xln⁡(sin⁡x)dx  Step 2: Use Integration by Parts on ∫cot⁡xln⁡(sin⁡x)dx  Let u=ln⁡(sin⁡x),dv=cot⁡x,du=cos⁡xsin⁡xdx,v=ln⁡(sin⁡x)  Step 3: Substitute the above into uv−∫vdu.  ln⁡(sin⁡x)2−∫ln⁡(sin⁡x)cos⁡xsin⁡xdx  Step 4: Use Integration by Substitution on ∫ln⁡(sin⁡x)cos⁡xsin⁡xdx.  ∫ln⁡uudu  Step 6: Use Integration by Substitution.  Let w=ln⁡u,dw=1udu  Step 7: Using w and dw above, rewrite ∫ln⁡uudu.  ∫wdw  Step 8: Use Power Rule: ∫xndx=xn+1n+1+C.  w22  Step 9: Substitute w=ln⁡u back into the original integral.  ln⁡u22  Step 10: Substitute u=sin⁡x back into the original integral.  ln⁡(sin⁡x)22  Step 11: Rewrite the integral with the completed substitution.  ln⁡(sin⁡x)22  Step 12: Add constant.  ln⁡(sin⁡x)22+C

Evaluate the indefinite integral. \int (\cot x)\ln (\sin x)dx

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